# Geodesics Orbiting a Singularity

Daniel Grieser  Mathematisches Institut, Universität Oldenburg
<EMAIL_ADDRESS>and  Jørgen Olsen Lye  Institut für
Differentialgeometrie, Leibniz Universität Hannover<EMAIL_ADDRESS>hannover.de

###### Abstract.

We study the behaviour of geodesics on a Riemannian manifold near a
generalized conical or cuspidal singularity. We show that geodesics entering a
small neighbourhood of the singularity either hit the singularity or approach
it to a smallest distance $\delta$ and then move away from it, winding around
the singularity a number of times. We study the limiting behaviour $\delta\to
0$ in the second case. In the cuspidal case the number of windings goes to
infinity as $\delta\to 0$, and we compute the precise asymptotic behaviour of
this number. The asymptotics have explicitly given leading term determined by
the warping factor that describes the type of cuspidal singularity. We also
discuss in some detail the relation between differential and metric notions of
conical and cuspidal singularities.

###### 2020 Mathematics Subject Classification:

53C22, 53A99, 53D25
Key words and phrases: Geodesics, Conical Singularity, Cuspidal Singularity,
Singular Hamiltonian Systems.

###### Contents

  1. 1 Introduction
    1. 1.1 Setting
    2. 1.2 Results
    3. 1.3 Methods, outline of proofs
    4. 1.4 Related work and further remarks
  2. 2 Generalized Warped Product Geometry
  3. 3 Geodesic equations
    1. 3.1 Review of the Hamiltonian approach
    2. 3.2 Geodesic equations for generalized warped product metrics
    3. 3.3 The warped product case; Clairaut’s integral
    4. 3.4 Estimates for the deviation from the warped product case
  4. 4 Proofs of the main theorems
  5. 5 Discussion
    1. 5.1 The function $\mathfrak{F}$ and the constant $C_{f}$
    2. 5.2 Example of $\mathfrak{F}$ failing to exist
    3. 5.3 Why convexity?
  6. 6 Metrics on isolated singularities
    1. 6.1 Singularities of type $[s]$
      1. 6.1.1 The cuspidal case
      2. 6.1.2 Conical singularities
    2. 6.2 Metrics on singularities of type $[s]$
      1. 6.2.1 Warped products
      2. 6.2.2 Generalized warped products
    3. 6.3 Summary of Section 6
  7. 7 Acknowledgements

##  1\. Introduction

Geodesics in Riemannian manifolds are among the most fundamental objects of
differential geometry. Besides their intrinsic interest as locally shortest
curves, or as trajectories of a free particle, they are used to define normal
coordinates, which are of great utility. Also, they play an essential role in
studying solutions of the wave equation, whose singularities propagate along
geodesics (see [10], for instance).

In this paper we study the behaviour of geodesics near an isolated conical or
cuspidal singularity of a Riemannian space. The precise definition of such
singularities is given below, but for a first illustration of our results
consider the example of the surface in $\mathbb{R}^{3}$ generated by rotating
the curve $x=z^{2}$, $z\geq 0$, around the $z$-axis, see the bottom left
picture in Figure 1. It has a cuspidal singularity at the origin $p=0$.

Figure 1. Illustrations of Theorem C for a conical metric $f(r)=r$ (above) and
a cuspidal metric $f(r)=r^{2}$ (below). The right hand pictures show the
product space $(0,R)\times Y$ while the left hand pictures illustrate the
geometric situation. In all cases, $R=1.5$ and $\delta=0.3$. The $r$ resp. $z$
direction is upwards. Only the downward moving part of the geodesic is shown
in the cuspidal case.

Consider the geodesics entering the neighbourhood $U=\\{z<1\\}$ of $p$ at any
given point. It is quite obvious that one of these will hit $p$ after a finite
time. Any other geodesic will move downward, reach a lowest point at
$z=\delta$ say, then move up again, and finally leave $U$ (this can be seen
using the classical Clairaut integral, for example, but we will prove it in
greater generality). Inside $U$ this geodesic will wind around $p$ a number of
times, as illustrated in Figure 1. In this special case, our results (see
Theorem C) imply that the number of windings is asymptotic to
$\frac{\pi}{C}\delta^{-1}$ as $\delta\to 0$, with
$C=\int_{-\pi/2}^{\pi/2}\sqrt{\cos(\vartheta)}\,d\vartheta\approx 2.4$.

Our setting and results are more general than this example in the following
ways: we allow more general profile functions $x=f_{0}(z)$ instead of $z^{2}$,
including $f_{0}(z)=z$ and $f_{0}(z)=e^{-1/z}$, for example. We also consider
the natural generalization of rotation surfaces to product manifolds
$(0,R)\times Y$ in any dimension with warped product metrics, where the cross
section $Y$ is any closed Riemannian manifold (generalizing $Y=S^{1}$ in the
surface case, where $f_{0}$ roughly corresponds to the warping function), and
we also allow perturbations of such warped products. See below for the precise
setting.

Our Main Theorems A, B, C are stated in Subsection 1.2 below. They can be
roughly summarized as follows.

######  Theorem 1 (Rough summary of all results).

Consider a space with a conical or cuspidal singularity $p$, with cross
section $Y$. Any geodesic $\gamma$ in a neighbourhood of $p$ is either radial,
i.e. it hits the singularity, or it approaches $p$, to a smallest distance
$\delta$, and then moves away from $p$, while winding around $p$. For a
winding geodesic $\gamma$ with small $\delta$ we have:

  * •

The distance of $\gamma$ to $p$ behaves as for a radial geodesic up to an
error of order $\delta$.

  * •

The $Y$-component of $\gamma$ closely follows a geodesic in $Y$.

  * •

The length of this geodesic in $Y$ (generalizing the number of windings in the
case $Y=S^{1}$) is asymptotic (as $\delta\to 0$) to
$C_{f}/f^{\prime}(\delta)$, for a constant $C_{f}$ only depending on $f$.

###  1.1. Setting

We now give a rough definition of the spaces that we consider in this paper.
We refer to Section 2 for the precise definitions of a Riemannian space with
isolated conical or cuspidal singularity along with technical assumptions. The
key point is that the metric has generalized warped product form near the
singularity, which we define as follows.

######  Definition 2.

Let $Y$ be a compact manifold and $R>0$. A generalized warped product metric
on $(0,R)\times Y$ is a smooth Riemannian metric of the form, with $r$ the
coordinate on $(0,R)$,

(1) $g=dr^{2}+f(r)^{2}\,h_{r}\,,\qquad h_{r}=h(r,y)dy^{2}$

where $f:(0,R)\to(0,\infty)$ is continuously differentiable and $h_{r}$ is a
Riemannian metric111The notation $h(r,y)\,dy^{2}$ is supposed to be suggestive
of the coordinate representation of $h_{r}$, which is
$h_{ij}(r,y)\,dy^{i}dy^{j}$. We use the Einstein summation convention. on $Y$
for each $r\in[0,R)$, depending continuously differentiably on $r$. The
function $f$ is called the warping function. If $h$ does not depend on $r$
then $g$ is called a warped product metric.

Of course, in (1) the factor $f^{2}$ could be incorporated into $h$, but we
will be interested in the case where $f(r)\to 0$ as $r\to 0$ while $h$ stays
non-degenerate. In this case distances between points $(r,y)$ and
$(r^{\prime},y^{\prime})$ tend to zero as $r\to 0$, $r^{\prime}\to 0$, so
geometrically one may complete the metric at $r=0$ by adding a single point,
which we call the singularity. We call the resulting metric space including
the singularity a Riemannian space with isolated singularity. Figure 1
illustrates the idea.

######  Definition 3.

In the setting of Definition 2, assume

> the function $f$ extends to a continuously differentiable map
>
> $f:[0,R)\to[0,\infty)$
>
> which satisfies
>
> (2) $f(0)=0\,,\quad f\text{ convex}.$

If $f^{\prime}(0)>0$ then we speak of a conical singularity, while if
$f^{\prime}(0)=0$ then we speak of a cuspidal singularity.

######  Remark 4.

The convexity of $f$ is not needed in the conical case if $f$ extends to a
$C^{2}$-function $f:[0,R)\to[0,\infty)$. See Remark 8 (a)-(c) below. See
Remark 8 (d) for some typical examples of warping functions covered by this
work.

###  1.2. Results

We give a detailed description of the geodesics in the pointed neighbourhood
$(0,R)\times Y$ of the conical/cuspidal singularity. We write geodesics as

$\gamma:I\to(0,R)\times Y\,,\qquad\gamma(t)=(r(t),y(t))\,,$

$I\subset\mathbb{R}$ an interval, and assume unit speed,
$|\dot{\gamma}|_{g}\equiv 1$, i.e. $\dot{r}^{2}+f(r)^{2}|\dot{y}|^{2}\equiv
1$, where a dot always indicates the derivative with respect to $t$ and
$|\dot{y}|$ is the length with respect to $h_{r}$.

First, we have the following dichotomy.

######  Theorem A (Radial and winding geodesics).

Let $g$ be a generalized warped product metric on $(0,R)\times Y$ as in (1).
Then any geodesic is of one of the following types.

radial:

$y$ is constant, and $\dot{r}$ is constant equal to $1$ or $-1$.

winding:

$\dot{y}(t)\neq 0$ for all $t$.

If the warping function satisfies (2) and if $R>0$ is small enough, then any
maximal winding geodesic $\gamma:I\to(0,R)\times Y$ satisfies in addition:

  1. (a)

The function $t\mapsto r(t)$ is strictly convex.

  2. (b)

$\gamma$ has finite length, i.e. $I=(T_{-},T_{+})$ with
$T_{\pm}\in\mathbb{R}$.

  3. (c)

$\gamma$ enters and leaves at $r=R$, i.e. $r(t)\to R$ as $t\to T_{\pm}$.

In the warped product case, $t\mapsto y(t)$ is a time reparametrised geodesic
of $(Y,h)$.

In the sequel, all geodesics will be assumed to be maximal, i.e. their domain
$I$ cannot be enlarged.

We now give a more precise description of the winding geodesics
$\gamma=(r,y)$. By (a) and (c) above, $r$ assumes its minimum at a unique
time, which we may and will always take to be $t=0$. We write $\delta=r(0)$
for the minimum. Since geodesics are uniquely determined by their initial
point, we get a parametrisation

(3) $(0,R)\times SY\to\\{\text{winding
geodesics}\\}\,,\quad(\delta,y_{0},v_{0})\mapsto\gamma_{\delta,y_{0},v_{0}}$

where $SY=\\{(y,v)\in TY\,:\ |v|_{h_{0}}=1\\}$ is the unit tangent bundle and
$\gamma_{\delta,y_{0},v_{0}}$ is the maximal geodesic starting at
$(\delta,y_{0})$ in direction $(0,v_{0})\in T_{(\delta,y_{0})}((0,R)\times
Y)$.

The next two theorems describe the behaviour of winding geodesics as
$\delta\to 0$. Theorem B describes how the radial component $r_{\delta}$
behaves, including a comparison theorem. Theorem C describes the $Y$ component
$y_{\delta}$, in particular the asymptotics of its length.

######  Theorem B (Radial component of geodesics).

Let $(X,d)$ be a Riemannian space with an isolated conical or cuspidal
singularity, and assume $R$ is small enough so (8) is satisfied.

  1. (a)

Let $\gamma_{\delta}=(r_{\delta},y_{\delta}):I_{\delta}\to(0,R)\times Y$,
$\delta\in(0,R)$, be any family of unit speed geodesics, with
$\min_{t}r_{\delta}(t)=r_{\delta}(0)=\delta>0$, with maximal interval of
existence $I_{\delta}$.

As $\delta\to 0$ we have $I_{\delta}\to(-R,R)$ (i.e. the endpoints of
$I_{\delta}$ converge to $\pm R$) and

$r_{\delta}(t)\to|t|\text{ for all }t\in(-R,R)\,.$

  2. (b)

In the case of a warped product with convex warping function $f$, the
following comparison principle holds. If $\gamma$, $\overline{\gamma}$ are two
unit speed geodesics which both reach their lowest point at $t=0$, and if
$r(0)<\overline{r}(0)$, then $r(t)<\overline{r}(t)$ for all time $t$.

In fact, we have precise error estimates for the convergence in (a), see Lemma
15. Note that combining (a) and (b) we obtain

$r(t)>|t|\text{ for all }t$

in the warped product case, for any winding geodesic. This is not obvious even
for a conical surface.

######  Theorem C (Angular component of geodesics).

Assume the setup of Theorem B, and let
$\gamma_{\delta}=(r_{\delta},y_{\delta})$ be a family of geodesics as in part
(a) of that theorem. Assume that the warping function $f$ satisfies the non-
oscillation condition (7).

  1. (a)

The length of the $Y$-projection of $\gamma_{\delta}$,

$\ell(y_{\delta})\coloneqq\int_{I_{\delta}}|\dot{y}_{\delta}(t)|_{h}\,dt$

satisfies

(4) $\ell(y_{\delta})\sim\frac{C_{f}}{f^{\prime}(\delta)},$

where $\sim$ means that the quotient of the left hand side by the right hand
side tends to one as $\delta\to 0$ and

$C_{f}\coloneqq\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\mathfrak{F}\left(\frac{1}{\cos(\vartheta)}\right)\,d\vartheta$

with $\mathfrak{F}$ defined in (7). The constant $C_{f}$ satisfies

$2\leq C_{f}\leq\pi,$

with $C_{f}=\pi$ in the conical case and $C_{f}\to 2$ for $f(x)=x^{\alpha}$ as
$\alpha\to\infty$. The sharper the cusp, the smaller is $C_{f}$ (see (40) for
a precise statement).

  2. (b)

Suppose that $y_{\delta}(0)\to y_{0}$, $f(\delta)\dot{y}_{\delta}(0)\to v_{0}$
as $\delta\to 0$ for some $(y_{0},v_{0})\in SY$. Let $\tilde{y}:\tilde{I}\to
Y$ be the unit speed geodesic in $(Y,h_{0})$ satisfying $\tilde{y}(0)=y_{0}$,
$\dot{\tilde{y}}(0)=v_{0}$, where

$\tilde{I}=(-\frac{\pi}{2},\frac{\pi}{2})\text{ in the conical case with
}f^{\prime}(0)=1,\quad\tilde{I}=\mathbb{R}\text{ in the cuspidal case.}$

Then $y_{\delta}$ converges after unit speed reparametrisation to $\tilde{y}$,
uniformly on compact subsets of $\tilde{I}$.

More explicitly, the statement in (b) is: Define the rescaled time $\tau$ by
$\tau(0)=0$ and

(5) $\frac{d\tau}{dt}=|\dot{y}_{\delta}|$

Then for $\tilde{I}_{\delta}=\tau(I_{\delta})$ and
$\tilde{y}_{\delta}:\tilde{I}_{\delta}\to Y$ defined by
$\tilde{y}_{\delta}(\tau(t))=y_{\delta}(t)$ we have, as $\delta\to 0$,

$\tilde{I}_{\delta}\to\tilde{I}\,,\quad\text{and}\quad\tilde{y}_{\delta}\to\tilde{y}\text{
uniformly on compact subsets of $\tilde{I}$. }$

The asymptotic length in (4) behaves as follows, as $\delta\to 0$:

$\frac{C_{f}}{f^{\prime}(\delta)}\to\frac{\pi}{f^{\prime}(0)}\ \text{ (conical
case)},\quad\frac{C_{f}}{f^{\prime}(\delta)}\to\infty\ \text{ (cuspidal
case).}$

This shows that parts (a) and (b) of the theorem are consistent, since
$|\tilde{I}_{\delta}|=\ell(\gamma_{\delta})$.

In particular, Theorem C(b) implies that every compact segment of a geodesic
on $Y$ (having length less than $\pi/f^{\prime}(0)$ in the conical case) is
the uniform limit of the $Y$-parts of some family $(\gamma_{\delta})$ of
geodesics on $X$.

######  Remark 5.

  1. (1)

It is worth stressing that the constant on the right hand side of (4) only
depends on $f$, and not on $h$, the dimension, or any detailed description of
$\gamma_{\delta}$. We discuss the function $\mathfrak{F}$ and the constant
$C_{f}$ in more detail in Section 5, including bounds and several examples.

  2. (2)

In the case of $Y=S^{1}$ of length $2\pi$ Equation (4) means that a geodesic
which almost hits the singularity winds roughly
$\frac{C_{f}}{2\pi}\cdot\frac{1}{f^{\prime}(\delta)}$ times around the
singularity before leaving. See Figure 1 for a couple of illustrations.

  3. (3)

The assumed convergence of the initial conditions of $y_{\delta}$ in part (b)
of Theorem C is always satisfied after passing to a subsequence, by
compactness.

###  1.3. Methods, outline of proofs

We formulate the geodesic equations as a Hamiltonian system of first order
ODEs. In the warped product case, the equations for $r$ completely decouple
and depend only on $f$. The motion in the $r$-direction then determines the
speed in the $Y$-direction via the unit speed condition. This allows us to
study the asymptotic behaviour quite explicitly. In the generalized warped
case, the equations of motion almost decouple. In particular, the leading
order behaviour turns out to be the same as in the warped case, and we derive
bounds which imply that the generalized warped case has the same asymptotic
behaviour as the warped one.

Perhaps the most surprising of our results is the length asymptotics (4). Here
is a quick rough explanation for the appearance of the term
$f^{\prime}(\delta)$, in the case of a cuspidal warped product metric: Denote
by $\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$ the angle between $\dot{\gamma}$
and the ‘circle’ of latitude $r=$ const, for a given geodesic
$\gamma=\gamma_{\delta}$. Then $|\dot{\gamma}|=1$ implies
$\cos\theta=f(r)|\dot{y}|$, $\sin\theta=\dot{r}$. The well-known Clairaut
relation (which we rederive) says that $f(r)\cos\theta$ is constant along
$\gamma$. Differentiating it in $t$ we obtain
$\dot{\theta}=f^{\prime}(r)|\dot{y}|$, or
$|\dot{y}|dt=\frac{1}{f^{\prime}(r)}d\theta$. The length of $\gamma$ is
obtained by integrating this, and (4) essentially follows from the fact that
$r$ is on the order of $\delta$ when $\theta$ is bounded away from $\pi/2$.

The structure of the paper is as follows. We start by describing the geometry
of the spaces we work with in some more detail in Section 2. We proceed by
stating the geodesic equations as a Hamiltonian system in Section 3. Here we
also introduce several useful variables and deduce a Clairaut-like relation.
Section 4 is the heart of the paper, where we analyse the equations of motion
and prove Theorem A, B and C. Section 5 deals with the constant $C_{f}$
appearing in the length asymptotics (4), including explicit computations of it
and the necessity of assuming convexity and that (7) exists. Section 6 relates
our notion of conical/cuspidal spaces to metric-free definitions of such
spaces. Here we also present some families of spaces for which our assumptions
hold.

###  1.4. Related work and further remarks

Our results are new only in the cuspidal case but give a unified treatment of
conical and cuspidal case. The conical case was treated by Melrose and Wunsch
in [14, Definition 1.4, Lemma 1.5] in the context of wave propagation, after
early work by Stone [16]. The first author discusses conical metrics in detail
in [8].

The family of geodesics hitting the singularity was analysed for a more
general class of $k$-cuspidal (i.e. $f(r)=r^{k}$, $k\geq 2$) metrics – those
arising from differential $k$-cuspidal singularities, see Subsection 6.2 and
in particular (48) – by Grandjean and the first author [6]. By Theorem A, the
geodesics hitting the singularity are simply given by constant $y$, for
generalized warped product metrics. In [6] it is shown that metrics on
$k$-cuspidal singularities are not of generalized warped product form (see
Subsection 6.2 and in particular (45) in the present work), and that the
family of geodesics hitting the singularity has a more complicated structure
in this setting.

Previously, Bernig and Lytchak [1] obtained first order information for
geodesics on general real algebraic sets $X\subset\mathbb{R}^{n}$, by showing
that any geodesic reaching a singular point $p$ of $X$ in finite time must
have a limit direction at $p$.

For more on warped products and geodesics, one can consult [3, Section 7] and
[15, Chapter 7].

##  2\. Generalized Warped Product Geometry

We start by being a bit more precise about the family of metrics $h_{r}$ in
our definition of generalized warped products.

######  Assumption 6 (Regularity assumption).

The metrics $h_{r}$ in the generalized warped product metric of Definition 2
are uniformly positive definite and depend uniformly $C^{1}$ on $r$. These
uniformity conditions are equivalent to

(6) $-2ch\leq\partial_{r}h\leq 2ch$

for all $r\in(0,R)$, for some constant $c\geq 0$.

This implies that the family extends to a continuous family of metrics on
$r\in[0,R]$.

Next, we introduce singular Riemannian spaces whose singularities are modelled
on generalized warped products as defined above.

######  Definition 7.

A metric space $(X,d)$ is called a Riemannian space with an isolated conical
or cuspidal singularity at $p\in X$ if

  * •

$X\setminus\\{p\\}$ is a smooth manifold, and the metric $d$ is induced by a
Riemannian metric $g_{X}$ on $X\setminus\\{p\\}$ ;

  * •

there is a neighbourhood $U$ of $p$ in $X$, a number $R>0$ and a compact
manifold $Y$, and a continuous map

$\beta:[0,R)\times Y\to U$

which sends $\\{0\\}\times Y$ to $p$ and restricts to an isometry
$\bigl{(}(0,R)\times Y,g\bigr{)}\to\bigl{(}U\setminus\\{p\\},g_{X}\bigr{)}$
where $g$ is a generalized warped product metric as in (1) and where $f$
satisfies the conditions of Definition 3. As in that definition, if
$f^{\prime}(0)>0$ then we speak of a conical singularity, while if
$f^{\prime}(0)=0$ then we speak of a cuspidal singularity.

See Section 6 for examples, in particular for the proof that surfaces of
revolution as indicated above fit into this framework, and for the relation of
this notion of conical/cuspidal singularity to other natural such notions.

######  Remark 8.

  1. (a)

As remarked above, the warping function $f$ is not uniquely determined by the
metric $g$: Replacing $f$ by $af$ and $h$ by $a^{-2}h$, where $a$ is positive
and $C^{1}$ on $[0,R)$, yields the same metric. This is the only freedom, by
the conditions on $h$. Thus, if we call two warping functions equivalent if
they differ by such a factor $a$ then $g$ determines the equivalence class
$[f]$ of $f$, and it makes sense to speak of a singularity of type $[f]$.

  2. (b)

If $f$ vanishes to finite order $k$ at zero and is $C^{k+1}$ on $[0,R)$ then
it is equivalent to $f_{k}(r)=r^{k}$.

  3. (c)

We use convexity of $f$ in several arguments. However, if $f$ is non-convex
but there is an equivalent function $af$ which is convex then it follows that
Theorems A, B(a) hold verbatim and Theorem C holds with $f$ replaced by $af$.
For instance, this is true in the $C^{2}$ conical case. Note that convexity of
$f$ implies $f^{\prime}(r)>0$ for $r>0$.

The example of Section 5.3 demonstrates that the asymptotics of Theorem C can
change without convexity. Convexity of $f$ implies that the angle $\theta$
between a geodesic and the level sets $\\{r\\}\times Y$ increases in $r$, see
(20). See also Remark 12 for an interpretation of the convexity of $f$ in
terms of curvature.

  4. (d)

Some classes of functions $f$ to have in mind are222The first and second
family overlap when $\alpha=1=\mu$. All three families are convex for small
values of $x$. More precisely, the second family is convex as long as
$y=\ln\left(\frac{1}{x}\right)$ satisfies $\mu y^{\mu}-y-(\mu-1)\geq 0$. The
third family is convex when $\sqrt[\beta]{\frac{\beta}{\beta+1}}\geq x$. Both
can be achieved by shrinking $R$.

     * •

$f(x)=x^{\alpha}$ for $\alpha\in[1,\infty)$.

     * •

$f(x)=\exp\left(-\ln\left(\frac{1}{x}\right)^{\mu}\right)$ for
$\mu\in[1,\infty)$.

     * •

$f(x)=\exp\left(-\frac{1}{x^{\beta}}\right)$ for $\beta\in(0,\infty)$.

We will identify $U\setminus\\{p\\}$ with $(0,R)\times Y$. The map $\beta$
should be thought of as a generalization of polar coordinates: For
$X=\mathbb{R}^{2}$ the map

$\beta:[0,\infty)\times
S^{1}\to\mathbb{R}^{2}\,,\quad(r,\phi)\mapsto(r\cos\phi,r\sin\phi)$

is an isometry over $r>0$ for the metric $dr^{2}+r^{2}d\phi^{2}$ on
$(0,\infty)\times S^{1}$ (where $S^{1}=\mathbb{R}/2\pi\mathbb{Z}$) and the
Euclidean metric on $\mathbb{R}^{2}\setminus\\{0\\}$. So $0\in\mathbb{R}^{2}$
can be considered a conical singularity in this sense, and the same is true
for any smooth point of a Riemannian manifold (with $Y$ equal to the sphere).
The space $[0,\infty)\times Y$ is sometimes called the blow-up of $X$ in $p$,
and the map $\beta$ the blow-down map.

Note that, while $g$ is a Riemannian metric in $r>0$, it is only positive
semi-definite at $r=0$, with any two points at $r=0$ having distance zero with
respect to $g$. This reflects the fact that the map $\beta$ crunches all
points at $r=0$ to the single point $p$. The warping function determines the
’speed’ at which the crunching happens as $r\to 0$. Also, note that the form
(1) of the metric implies that $r$ is the distance to the singularity $p$.

######  Remark 9.

There is no unique answer to the question what the ’correct’ definition of a
Riemannian space with an isolated singularity is. While our notion of
singularity is quite general in terms of $f$ and $h_{r}$, it is somewhat
restrictive in requiring that there are no mixed terms in the metric (1), i.e.
that coordinates $r,y$ can be chosen so that the lines $y=$ const are
perpendicular to the hypersurfaces $r=$ const. See Section 6.2.2 for a natural
example where this is not satisfied. We leave it for future work to analyse
the geodesic flow for some of these more general metrics.

For Theorem C, we need $f$ to satisfy the following condition:

######  Assumption 10 (Non-oscillation condition).

For the inverse function $F\coloneqq f^{-1}$ the limit

(7) $\mathfrak{F}(\sigma)\coloneqq\lim_{\epsilon\to
0}\frac{F^{\prime}(\sigma\epsilon)}{F^{\prime}(\epsilon)}$

exists for $\sigma\in[1,\infty)$, and the convergence is uniform on compact
subsets.

This condition is satisfied for the examples above. See Lemma 19 for a family
of cuspidal examples where it is not satisfied. Furthermore, we need (except
in Section 3) to limit the variation of $h_{r}$ over the interval $(0,R)$:

######  Assumption 11 (Small perturbation condition).

The constant $c$ in (6) can be chosen so that

(8) $Rc<1.$

Of course this can be achieved by shrinking $R$. The condition (8) is used to
prove (20) below, which in turn gets used implicitly several times.

######  Remark 12.

The small perturbation condition along with the convexity of $f$ has an
interpretation in terms of the mean curvature of the level sets of $r$: A
standard computation shows that the mean curvature vector of $\\{r\\}\times
Y\subset(0,R)\times Y$ is $H=H_{0}\partial_{r}$ where

$H_{0}=-\dim(Y)\frac{f^{\prime}}{f}-\frac{1}{2}\text{Tr}_{h}(\partial_{r}h)\,.$

By the bound (6), we can bound $H_{0}$ as

$-\dim(Y)\left(\frac{f^{\prime}}{f}+c\right)\leq
H_{0}\leq-\dim(Y)\left(\frac{f^{\prime}}{f}-c\right)$

The convexity of $f$ and $f(0)=0$ imply
$\frac{f^{\prime}(r)}{f(r)}\geq\frac{1}{r}\geq\frac{1}{R}$ (see Lemma 14), so
the small perturbation condition implies a negative upper bound for the scalar
mean curvature,

$H_{0}\leq-\frac{\dim(Y)}{R}(1-Rc)<0.$

We will not use this geometric interpretation directly.

##  3\. Geodesic equations

In this section we analyse the geodesic equations for generalized warped
product metrics of the type (1), but without extra conditions on the warping
function $f$.

###  3.1. Review of the Hamiltonian approach

We use the Hamiltonian description of the geodesic flow. Recall that this
means that (constant speed) geodesics on a Riemannian manifold $(Y,h)$ are the
projections to $Y$ of curves in the cotangent bundle $T^{*}Y$, which are the
integral curves of the Hamiltonian vector field for the Hamilton function
$\mathcal{H}_{h}:T^{*}Y\to\mathbb{R}$,

$\mathcal{H}_{h}(y,\eta)=\frac{1}{2}|\eta|^{2}_{h_{y}}\,,\quad\eta\in
T^{*}_{y}Y$

where by $h_{y}$ we also denote the metric on $T^{*}_{y}Y$ dual to the metric
$h_{y}$ on $T_{y}Y$. In coordinates,
$|\eta|^{2}_{h_{y}}=h^{ij}(y)\eta_{i}\eta_{j}$ where $(h^{ij})$ is the inverse
matrix of $(h_{ij})$. Often we write simply $|\eta|_{h}$ or $|\eta|$. The
Hamilton vector field is, in coordinates,

$\mathcal{X}_{h}=\frac{\partial\mathcal{H}_{h}}{\partial\eta}\partial_{y}-\frac{\partial\mathcal{H}_{h}}{\partial
y}\partial_{\eta}$

so its integral curves, also called lifted geodesics, are solutions
$t\mapsto(y(t),\eta(t))$ of the system of differential equations

$\dot{y}=\frac{\partial\mathcal{H}_{h}}{\partial\eta}(y,\eta)\qquad\dot{\eta}=-\frac{\partial\mathcal{H}_{h}}{\partial
y}(y,\eta)\,.$

Geodesics are then the $y(t)$ parts of solutions of this system. It is a basic
fact that the Hamiltonian function is constant along integral curves, i.e.
$\mathcal{H}_{h}(y(t),\eta(t))=$ const. Note that

$\frac{\partial\mathcal{H}_{h}}{\partial\eta}(y,\eta)=\eta^{\sharp}$

where $\eta\mapsto\eta^{\sharp}$, $T^{*}Y\to TY$ is the isomorphism induced by
the metric. In coordinates, $(\eta^{\sharp})^{i}=h^{ij}\eta_{j}$. So for a
geodesic $t\mapsto y(t)$ its lift is the curve $(y(t),\eta(t))$ where
$\eta^{\sharp}(t)=\dot{y}(t)$, and this explains the relationship of the
Hamiltonian approach to geodesics (with lift in the cotangent bundle) to the
more standard approach where the lift is the curve $(y(t),\dot{y}(t))$ in the
tangent bundle.

###  3.2. Geodesic equations for generalized warped product metrics

We apply this general discussion with $Y$ replaced by $(0,R)\times Y$, with
the metric $g$ in (1). Points are denoted $(r,y)$, and the dual cotangent
variables are denoted $(\xi,\eta)$, so $\xi\in\mathbb{R}$,
$\eta\in\mathbb{R}^{n-1}$ in coordinates. The Hamilton function is

(9)
$\mathcal{H}_{g}=\frac{1}{2}\left(\xi^{2}+\frac{|\eta|^{2}_{h}}{f(r)^{2}}\right)$

As mentioned above, $\mathcal{H}_{g}$ is constant along integral curves. We
will always consider unit speed geodesics, i.e. integral curves lying on the
hypersurface $\mathcal{H}_{g}=\frac{1}{2}$, or

(10) $1=\xi^{2}+\frac{|\eta|^{2}_{h}}{f(r)^{2}}.$

We calculate the Hamilton vector field for $\mathcal{H}_{g}$. Since
$\mathcal{H}_{g}$ depends on $y,\eta$ only through $|\eta|^{2}_{h}$ we have

(11)
$\mathcal{X}_{g}=\mathcal{X}_{\mathrm{rad}}+\frac{1}{f(r)^{2}}\mathcal{X}_{h}$

where
$\mathcal{X}_{\mathrm{rad}}=\frac{\partial\mathcal{H}_{g}}{\partial\xi}\partial_{r}-\frac{\partial\mathcal{H}_{g}}{\partial
r}\partial_{\xi}$ governs the radial motion and $\mathcal{X}_{h}$ is the
Hamilton vector field of $\mathcal{H}_{h}$, hence governs the motion in the
$Y$ directions. Recall that the latter depends parametrically on $r$ since $h$
does, and as such defines a family of vector fields on $Y$.

We first observe that $\mathcal{X}_{g}$ is tangential to the submanifold
$\\{\eta=0\\}$ since the $\partial_{\eta}$ coefficient, which is
$-f(r)^{-2}\partial|\eta|^{2}_{h_{y}}/\partial y$, vanishes at $\eta=0$. This
implies that any integral curve of $\mathcal{X}_{g}$ satisfies

either $\eta(t)=0$ for all $t$ or $\eta(t)\neq 0$ for all $t$.

The Hamilton equations for $\dot{r}$ and $\dot{y}$ are

$\dot{r}=\frac{\partial\mathcal{H}_{g}}{\partial\xi}=\xi\,,\quad\dot{y}=\frac{\partial\mathcal{H}_{g}}{\partial\eta}=\frac{1}{f(r)^{2}}\frac{\partial\mathcal{H}_{h}}{\partial\eta}=\frac{1}{f(r)^{2}}\eta^{\sharp}\,.$

From this we deduce the first part of Theorem A: Geodesics with $\eta\equiv 0$
have constant $y$, and $\dot{r}=\xi\equiv\pm 1$ by the unit speed condition
(10), so they are radial. All other geodesics have $\dot{y}(t)\neq 0$ for all
$t$, so they are winding.

We now consider winding geodesics, i.e. $\eta(t)\neq 0$ for all $t$, and
derive the full Hamilton equations for them. Calculating
$\frac{\partial\mathcal{H}_{g}}{\partial
r}=\frac{|\eta|^{2}_{h}}{f(r)^{2}}\left(-\frac{f^{\prime}(r)}{f(r)}+\frac{\partial_{r}|\eta|}{|\eta|}\right)$
and using (10) we get

(12)
$\mathcal{X}_{\mathrm{rad}}=\xi\partial_{r}+(1-\xi^{2})\left(\frac{f^{\prime}(r)}{f(r)}-\tfrac{\partial_{r}|\eta|}{|\eta|}\right)\partial_{\xi}$

We write $\tfrac{\partial_{r}|\eta|}{|\eta|}$ in small font to emphasize that
it is to be considered as a small perturbation (it vanishes in the warped
product case). Correspondingly, the lifted geodesics are solutions
$t\mapsto(r(t),y(t),\xi(t),\eta(t))$ of the system

(13) $\displaystyle\dot{r}$ $\displaystyle=\xi$ $\displaystyle\dot{\xi}$
$\displaystyle=(1-\xi^{2})\left(\frac{f^{\prime}(r)}{f(r)}-\tfrac{\partial_{r}|\eta|}{|\eta|}\right)$
(14) $\displaystyle\dot{y}$
$\displaystyle=f(r)^{-2}\,\partial_{\eta}\mathcal{H}_{h}$
$\displaystyle\dot{\eta}$
$\displaystyle=-f(r)^{-2}\,\partial_{y}\mathcal{H}_{h}$

By (10), (2), and the assumption $\eta(t)\neq 0$ for all $t$, the variable
$\xi$ is constrained to lie in $(-1,1)$, so we may introduce a new variable
$\theta$ (i.e. coordinate on the hypersurface $\mathcal{H}_{g}=\frac{1}{2}$)
by

$\xi=\sin\theta\,,\qquad\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\,.$

This will simplify the calculations below. The equations (13) then turn into

(13’) $\displaystyle\dot{r}$ $\displaystyle=\sin\theta$
$\displaystyle\dot{\theta}=\left(\frac{f^{\prime}(r)}{f(r)}-\tfrac{\partial_{r}|\eta|}{|\eta|}\right)\cos\theta$

and the unit speed condition (10) turns into

(15) $f(r)\cos\theta=|\eta|\,.$

Note that $\dot{r}=\sin\theta$, $|\dot{\gamma}|=1$ implies that $\theta$ is
the angle between $\dot{\gamma}$ and the ’circle of latitude’ $\\{r\\}\times
Y$.

###  3.3. The warped product case; Clairaut’s integral

In the warped product case, i.e. where $h$ does not depend on $r$, two
simplifications happen: The vector field $\mathcal{X}_{\mathrm{rad}}$ is
independent of $\eta$ since the term $\partial_{r}|\eta|$ vanishes, and the
vector field $\mathcal{X}_{h}$ is independent of $r$.

This means that lifted geodesics $t\mapsto(r(t),\theta(t),y(t),\eta(t))$ are
given as follows:

  1. (1)

$(r(t),\theta(t))$ solves the system (13’)

  2. (2)

$(y(t),\eta(t))$ is an integral curve for the Hamilton vector field
$\mathcal{X}_{h}$, but with time reparametrised using the factor $f(r)^{-2}$
with $r=r(t)$ obtained in step 1.

Here we use that the vector field $\mathcal{H}_{h}$ and the time-dependent
vector field $\frac{1}{f(r(t))^{2}}\mathcal{H}_{h}$ on $Y$ have the same
integral curves except for time reparametrisation. We also see that, along
each integral curve,

(16) $f(r)\cos\theta={\mathrm{const}}\,,$

either by direct calculation from (13’) or using (15) and the fact that
$|\eta|$ is constant along integral curves for $\mathcal{X}_{h}$. The relation
(16) completely determines the solutions of the $r,\theta$ system up to time
parametrisation.

The expression on the left of (16) is known as Clairaut’s integral in the
context of surfaces of revolution.

###  3.4. Estimates for the deviation from the warped product case

If $h$ depends on $r$ then the ‘Clairaut integral ’ $f(r)\cos\theta$ is not
constant along geodesics, but we can estimate its variation.

By (15) the Clairaut integral equals $|\eta|$, so we consider this quantity.
Note that $|\eta|=|\eta|_{h}$ depends on $r$ through $h$.

We recall that Assumption 6 says

(6’) $-2ch\leq\partial_{r}h\leq 2ch$

for a constant $c\geq 0$.

######  Lemma 13.

With $c$ as above, we have for any $\eta\in T^{*}Y\setminus\\{0\\}$:

(17) $\displaystyle-c\leq\tfrac{\partial_{r}|\eta|}{|\eta|}\leq c\,.$

Furthermore, along any lifted geodesic we have

(18) $\left|\frac{d}{dt}|\eta|\right|\leq c|\eta|\,|\sin\theta|$

Estimate (18) quantifies the fact that lifted geodesics are tangent to
$\\{\eta=0\\}$.

###### Proof.

Let $h^{-1}$ denote the inverse of $h$ considered as a matrix in some local
coordinates. We multiply (6’) by $h^{-1}$ from left and right to get
$-2ch^{-1}\leq\partial_{r}h^{-1}\leq 2ch^{-1}$ and therefore
$-2c|\eta|^{2}\leq\partial_{r}|\eta|^{2}\leq 2c|\eta|^{2}$. This implies (17).

Lifted geodesics are integral curves of $\mathcal{H}_{g}$, and along such a
curve we have

$\displaystyle\frac{d}{dt}\mathcal{H}_{h}$
$\displaystyle=(\partial_{y}\mathcal{H}_{h})\dot{y}+(\partial_{\eta}\mathcal{H}_{h})\dot{\eta}+(\partial_{r}\mathcal{H}_{h})\dot{r}$
$\displaystyle=(\partial_{r}\mathcal{H}_{h})\dot{r}$

using (14), and this gives $\frac{d}{dt}|\eta|=\partial_{r}|\eta|\sin\theta$
from which we deduce (18) using (17). ∎

##  4\. Proofs of the main theorems

In this section we use the results from the previous section in combination
with the properties (2) of $f$ to prove Theorems A, B, and C.

First, we supplement the estimates in Lemma 13 by estimates that use the
special properties of $f$.

######  Lemma 14.

A warping function $f$ as in (2) satisfies

(19) $\frac{f^{\prime}(r)}{f(r)}\geq\frac{1}{r}$

Furthermore, if $R<\frac{1}{c}$ (with $c$ satisfying (6)) then along any
lifted winding geodesic we have

(20) $\dot{\theta}>0$

###### Proof.

(19) follows from $f(0)=0$ and the convexity of $f$:
$f(r)=\int_{0}^{r}f^{\prime}(s)\,ds\leq rf^{\prime}(r)$.

The equation (13’) for $\dot{\theta}$ along with (19) and (17) imply
$\dot{\theta}\geq(\frac{1}{r}-c)\cos\theta$. Also $\cos\theta>0$ for winding
geodesics, which gives the second claim. ∎

###### Proof of Theorem A.

By the Clairaut-like relation (15), we see that $\cos\theta\neq 0$ for winding
geodesics. From $\dot{r}=\sin\theta$ and (20) we get

$\ddot{r}=\dot{\theta}\cos\theta>0\,,$

so $r$ is strictly convex. Then (b), (c) follow easily since geodesics exist
as long as $r<R$.

The statement about reparametrisation in the warped product case was explained
in Section 3.3. ∎

For the proof of Theorem B it is useful to introduce two new variables $\rho$
and $u$ (i.e. functions on $(0,R)\times Y$, resp. its cotangent bundle),
defined by the identities:

(21) $\displaystyle\rho$ $\displaystyle=f(r)$ $\displaystyle\rho\sin\theta$
$\displaystyle=f(u)$

We will use the following bounds on the variation of $\eta$ and $r$ along a
geodesic.

######  Lemma 15.

Assume $R<\frac{1}{c}$, with $c$ defined in (6). Let $\gamma_{\delta}$ be a
family of geodesics as in Theorem B(a). Then

(22) $f(\delta)e^{-c(r_{\delta}-\delta)}\leq|\eta_{\delta}|\leq
f(\delta)e^{c(r_{\delta}-\delta)}.$ (23) $(1-C\delta)\,|t|\leq
r_{\delta}(t)\leq|t|+\delta$

holds for all time $t$ and all $\delta$. Here $C=ce^{cR}$.

###### Proof of Lemma 15.

We will drop all $\delta$-subscripts, writing $r=r_{\delta}$ and so on. We may
assume $t>0$. Then $\theta>0$ since $\theta(0)=0$ and $\dot{\theta}>0$ by
(20).

We first prove (22). We rewrite (18) as
$\bigl{|}\frac{d}{dt}\log|\eta|\bigr{|}\leq c\sin\theta=c\dot{r}$, so

$-c\dot{r}\leq\frac{d}{dt}\log|\eta|\leq c\dot{r}\,.$

We integrate this from $t=0$ and exponentiate. Using $r(0)=\delta$ and
$|\eta(0)|=f(\delta)$ (from (15) since $\theta(0)=0$) we get (22).

We now prove (23). The unit speed condition implies $|\dot{r}(t)|\leq 1$, so

$r(t)-\delta=r(t)-r(0)=\int_{0}^{t}\dot{r}(s)\,ds\leq t.$

This gives the upper bound in (23)

For the lower bound we first consider the warped product case, for which
$c=0$, hence $C=0$, so we need to show

(24) $|t|\leq r_{\delta}(t).$

Differentiating $f(u)=f(r)\sin\theta$ in $t$ we get

(25)
$\dot{u}f^{\prime}(u)=\dot{r}f^{\prime}(r)\sin\theta+\dot{\theta}f(r)\cos\theta$

which by virtue of $\dot{r}=\sin\theta$,
$\dot{\theta}=\frac{f^{\prime}(r)}{f(r)}\cos\theta$ simplifies to

(26) $\dot{u}f^{\prime}(u)=f^{\prime}(r)$

Now $f(u)\leq f(r)$ by (21), so $u\leq r$ since $f$ is strictly increasing.
Then $f^{\prime}(u)\leq f^{\prime}(r)$ since $f^{\prime}$ is increasing. So
(26) implies

$\dot{u}\geq 1\,.$

From $u(0)=0$ we now get $t\leq u(t)\leq r(t)$ as claimed.

Now we prove the lower bound in (23) for the generalized warped product case.
In fact, we will prove the stronger lower bound

(27) $\left(1-C\frac{f(\delta)}{f^{\prime}(\delta)}\right)|t|\leq
r_{\delta}(t)$

which implies (23) by (19).

We use (25) again, but now we have the extra summand
$-\frac{\partial_{r}|\eta|}{|\eta|}\cos\theta$ in $\dot{\theta}$ (see (13’)),
which equals $-\frac{\partial_{r}|\eta|}{f(r)}$ by (15). Therefore, (26) is
replaced by

$\dot{u}f^{\prime}(u)=f^{\prime}(r)-\partial_{r}|\eta|\,\cos\theta=f^{\prime}(r)\left[1-\frac{\partial_{r}|\eta|}{f^{\prime}(r)}\cos\theta\right]$

This implies (27) by the same argument as in the warped case, provided

$1-\frac{\partial_{r}|\eta|}{f^{\prime}(r)}\cos\theta\geq
1-C\frac{f(\delta)}{f^{\prime}(\delta)}\,.$

Finally, this inequality follows from $f^{\prime}(r)\geq f^{\prime}(\delta)$
(since $r\geq\delta$ and $f^{\prime}$ is increasing), $\cos\theta\leq 1$, and
$\partial_{r}|\eta|\leq Cf(\delta)$. The latter inequality follows from (17)
and (22), with $C=ce^{cR}$. ∎

###### Proof of Theorem B.

By symmetry, it suffices to consider the $t>0$ part of the geodesics. Then
$\theta>0$ since $\theta(0)=0$ and $\dot{\theta}>0$. We write
$I_{\delta}=(T_{-}^{\delta},T_{+}^{\delta})$. By Theorem A part (b) and (c),
we have $r_{\delta}(T_{\pm}^{\delta})\coloneqq\lim_{t\to
T_{\pm}^{\delta}}r_{\delta}(t)=R$. By Lemma 15, we therefore have

$(1-C\delta)|T_{\pm}^{\delta}|\leq R\leq|T_{\pm}^{\delta}|+\delta,$

and sending $\delta\to 0$ gives

$T_{\pm}^{\delta}\xrightarrow{\delta\to 0}\pm R.$

Also $r_{\delta}(t)\to|t|$ follows from (23).

It remains to prove (b) in the warped product case. So let $\gamma=(r,y)$ and
$\overline{\gamma}=(\overline{r},\overline{y})$ be two geodesics with
$\delta=r(0)$, $\overline{\delta}=\overline{r}(0)$ and
$\delta<\overline{\delta}$. We write $u$ and $\overline{u}$ for the function
defined in (21) associated to $r$ and $\overline{r}$ respectively. By the
Clairaut-like relation (16) and the definition of $u$ (21) we have

(28) $f(r)^{2}=f(u)^{2}+f(\delta)^{2}$

and similarly for $f(\overline{r})$. Since $\partial_{r}h=0$, we may use (26)
to find

(29) $\frac{d}{dt}f(u)=f^{\prime}(r).$

and similarly for $f(\overline{u})$. The convexity of $f$ then implies

$\frac{d}{dt}\left(f(\overline{u})-f(u)\right)\geq 0$

for all $t$ such that $\overline{r}(t)\geq r(t)$. Assume for contradiction
that there is a maximal $t_{0}<T_{\delta}$ such that $\overline{r}(t)>r(t)$ on
$[0,t_{0})$. By continuity, $r(t_{0})=\overline{r}(t_{0})$. By (29), we
therefore have

$f(\overline{u}(t_{0}))-f(u(t_{0}))=\int_{0}^{t_{0}}f^{\prime}(\overline{r}(t))-f^{\prime}(r(t))\,dt\geq
0.$

Inserting this into (28), we find

$f(\overline{r}(t_{0}))^{2}-f(\overline{\delta})^{2}=f(\overline{u}(t_{0}))^{2}\geq
f(u(t_{0}))^{2}=f(r(t_{0}))^{2}-f(\delta)^{2}$

which rearranges to yield

$f(\overline{r}(t_{0}))^{2}-f(r(t_{0}))^{2}\geq
f(\overline{\delta})^{2}-f(\delta)^{2}>0.$

This shows $\overline{r}(t_{0})>r(t_{0})$, contradicting the maximality of
$t_{0}$.

∎

######  Remark 16.

The proof of the comparison theorem might seem a bit indirect, using $f(u(t))$
instead of $r(t)$. The reason being that it is in general false that
$\overline{r}(t)-r(t)$ is increasing. Indeed, for the conical case $f(r)=r$,
one can solve the equation of motion explicitly to find
$r(t)=\sqrt{t^{2}+\delta^{2}}$, and $\overline{r}(t)-r(t)$ is strictly
decreasing but never 0 when $\overline{\delta}>\delta$. The auxiliary function
$f(u(t))$ is in this conical case simply $f(u(t))=t=f(\overline{u}(t)$, so
$f(\overline{u}(t))-f(u(t))=0$ is increasing.

###### Proof of the first part of Theorem C.

We will use the coordinates $\rho=f(r)$ and $\sin\theta=\xi$ again. We will
drop the $\delta$-subscripts on all the variables. Since (20) says
$\dot{\theta}>0$, we can and will parametrise using $\theta$ instead of $t$.
From the equation of motion $\eqref{eqn:ham phi eta}$, and the Clairaut-like
relation (16), we have

(30) $|\dot{y}|=\frac{\cos(\theta)}{\rho}$

(we write $|\dot{y}|=|\dot{y}|_{h}:=|\dot{y}|_{h_{r_{\delta}(t)}}$).

Recall that $F$ is the inverse function of $f$. Using the equation of motion
(13’), $\partial_{r}=f^{\prime}(r)\partial_{\rho}$ and
$\frac{1}{f^{\prime}(r)}=F^{\prime}(\rho)$ we can write the measure as

(31)
$|\dot{y}|dt=\frac{F^{\prime}(\rho)}{1-\rho\partial_{\rho}\log|\eta|)}\,d\theta\,.$

We again start by assuming $h$ does not depend on $r$, since the argument
becomes more transparent in this case. In this case the formula (31)
simplifies to the elegant

$|\dot{y}|dt=F^{\prime}(\rho)\,d\theta,$

and one can use the Clairaut-like relation (16) to find

$\rho=\frac{\rho_{0}}{\cos(\theta)},$

where $\rho_{0}\coloneqq f(\delta)$. Hence

(32)
$f^{\prime}(\delta)|\dot{y}|dt=\frac{1}{F^{\prime}(\rho_{0})}|\dot{y}|dt=\cfrac{F^{\prime}\left(\cfrac{\rho_{0}}{\cos\theta}\right)}{F^{\prime}(\rho_{0})}\,d\theta.$

When $f$ is convex, $F^{\prime}$ is decreasing, and the right hand integrand
is bounded by $1$. So by the dominated convergence theorem and the non-
oscillation condition (7),

$\lim_{\delta\to
0}f^{\prime}(\delta)\int_{T_{-}}^{T_{+}}|\dot{y}|dt=\int_{-\pi/2}^{\pi/2}\lim_{\rho_{0}\to
0}\cfrac{F^{\prime}\left(\cfrac{\rho_{0}}{\cos\theta}\right)}{F^{\prime}(\rho_{0})}\,d\theta=\int_{-\pi/2}^{\pi/2}\mathfrak{F}\left(\frac{1}{\cos\theta}\right)\,d\theta=C_{f}.$

This proves part (a) when $h$ is independent of $r$. For the general case, we
return to (31). Using the bounds (17) again results in

$\frac{F^{\prime}(\rho)}{1+c\rho
F^{\prime}(\rho)}\,d\theta\leq|\dot{y}|dt\leq\frac{F^{\prime}(\rho)}{1-c\rho
F^{\prime}(\rho)}\,d\theta\,.$

Since $F$ is concave and $F(0)=0$, we have $\rho F^{\prime}(\rho)\leq
F(\rho)$, thus

(33)
$\frac{F^{\prime}(\rho)}{1+cF(\rho)}\,d\theta\leq|\dot{y}|dt\leq\frac{F^{\prime}(\rho)}{1-cF(\rho)}\,d\theta\,.$

We therefore need bounds on both sides of (33). Recall that $\rho$ is a
function of $\delta$ and $\theta$. We consider its behaviour as $\delta\to 0$
for fixed $\theta$. The bounds (22) say, since $|\eta|=\rho\cos\theta$,

(34)
$\frac{f(\delta)}{\cos\theta}e^{-c({r}-\delta)}\leq{\rho}\leq\frac{f(\delta)}{\cos\theta}e^{c({r}-\delta)},$

so

$\lim_{\delta\to 0}{\rho(\theta)}=0$

for every $\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$. But then also
${r}=F({\rho})\xrightarrow{\delta\to 0}0$. Hence, using (34) and
$f(\delta)=\rho_{0}$ we get

$\lim_{\delta\to 0}\frac{{\rho}}{\rho_{0}}\cos\theta=1.$

Since, per assumption,

$\lim_{\epsilon\to
0}\frac{F^{\prime}(\epsilon\sigma)}{F^{\prime}(\epsilon)}=\mathfrak{F}(\sigma)$

and the convergence is uniform on compacts, we have

$\lim_{\delta\to 0}\frac{F^{\prime}({\rho})}{F^{\prime}(\rho_{0})(1\pm
cF({\rho}))}=\lim_{\delta\to
0}\frac{F^{\prime}\left(\frac{{\rho}}{\rho_{0}}\cos\theta\cdot\frac{\rho_{0}}{\cos\theta}\right)}{F^{\prime}(\rho_{0})}=\mathfrak{F}\left(\frac{1}{\cos\theta}\right).$

Both the upper and lower bounds in (33) are bounded functions, hence
integrable. By the dominated convergence theorem again, we find

$\int_{-\pi/2}^{\pi/2}\mathfrak{F}\left(\frac{1}{\cos\theta}\right)\,d\theta\leq\lim_{\delta\to
0}f^{\prime}(\delta)\ell(y_{\delta})\leq\int_{-\pi/2}^{\pi/2}\mathfrak{F}\left(\frac{1}{\cos\theta}\right)\,d\theta.$

This proves part (a) in general. ∎

We now turn to the second part of Theorem C. We will rescale the time $\tau$
according to (5),

$\frac{d\tau}{dt}=|\dot{y}_{\delta}|=\frac{|\eta|}{f(r(t))^{2}}.$

subject to $\tau(0)=0$. We also introduce the rescaled angular momentum
variable

(35) $\overline{\eta}(t)\coloneqq\frac{\eta(t)}{f(\delta)},$

which by (14) implies $\overline{\eta}^{\\#}(0)=f(\delta)\dot{y}(0)$. We
recall our convention for the time-rescaled variables,

$\displaystyle\tilde{y}(\tau(t))$ $\displaystyle\coloneqq y(t),$
$\displaystyle\tilde{\eta}(\tau(t))\coloneqq\overline{\eta}(t).$

For most of the proof, we will not write out the $\delta$-subscript on
$y,\tilde{y},\eta,\tilde{\eta}$. We will use a slash ′ for $\tau$-derivatives
and a dot $\dot{}$ for $t$-derivatives. Recall that we assume
$\tilde{\eta}(0)=\overline{\eta}(0)=f(\delta)\dot{y}(0)\to v_{0}$ as
$\delta\to 0$.

###### Proof of the second part of Theorem C.

By the unit speed parametrisation, the length of the time interval is the same
as the length of the $Y$-component of the geodesic

$\Delta\tau=\int_{T_{-}^{\delta}}^{T_{+}^{\delta}}\frac{d\tau}{dt}\,dt=\ell(y_{\delta}).$

By the first part of Theorem C, this has the asymptotic behaviour

(36) $\Delta\tau\sim\frac{C_{f}}{f^{\prime}(\delta)}.$

The equations of motion (14) get rescaled by $\frac{dt}{d\tau}$ and read

(37) $\displaystyle\tilde{\eta}^{\prime}(\tau)$
$\displaystyle=\frac{\dot{\eta}(t)}{f(\delta)|\dot{y}|}=-\frac{\partial_{y}|\tilde{\eta}|^{2}}{2|\tilde{\eta}|}\quad\qquad\tilde{y}^{\prime}=\frac{\tilde{\eta}^{\sharp}}{|\tilde{\eta}^{\sharp}|}\,.$

We stress that $\tilde{\eta}$ is both a rescaled and time-reparametrised
quantity. We first observe that the bound (22) tells us

$C^{\prime}\leq|\tilde{\eta}|\leq C$

uniformly in $\tau$ and $\delta$, where $C^{\prime}>0$. Furthermore, since $h$
is $C^{1}$ in the $y$-directions and uniformly non-degenerate down to $r=0$,
we can find a constant $\tilde{c}\geq 0$ such that

$\left|\partial_{y}|\tilde{\eta}|^{2}\right|\leq
2\tilde{c}|\tilde{\eta}|^{2},$

hence, by the unit speed condition and equation of motion (37) respectively,

$d(\tilde{y}_{\delta}(\tau),\tilde{y}_{\delta}(\sigma))\leq\int_{\sigma}^{\tau}\left|\tilde{y}_{\delta}^{\prime}(\upsilon)\right|\,d\upsilon\leq|\tau-\sigma|,$

and

$|\tilde{\eta}_{\delta}(\tau)-\tilde{\eta}_{\delta}(\sigma)|=\left|\int_{\sigma}^{\tau}\tilde{\eta}_{\delta}^{\prime}(\upsilon)\,d\upsilon\right|\leq\tilde{C}|\tau-\sigma|.$

This shows that the families $\tilde{y}_{\delta},\tilde{\eta}_{\delta}$ are
uniformly (in $\tau$ and $\delta$) Lipschitz continuous. Let
$K\subset\tilde{I}_{\delta}$ be any compact subset. By the Arzelà-Ascoli
theorem, there is a subsequence $\delta_{n}\to 0$ such that
$(\tilde{y}_{\delta_{n}},\tilde{\eta}_{\delta_{n}})\to(\tilde{y},\tilde{\eta})$
in $C(K;T^{*}Y)$. Since we can rewrite the differential equations (37) as
integral equations, the limit $(\tilde{y},\tilde{\eta})$ satisfies the same
equations, with respect to the metric $h_{0}$ and with initial condition
$\tilde{y}(0)=y_{0}$ and $\tilde{\eta}(0)=v_{0}$. By the same argument, any
subsequence of $(\tilde{y}_{\delta},\tilde{\eta}_{\delta})$ has a convergent
subsequence, whose limit must be the same $(\tilde{y},\tilde{\eta})$ since it
satisfies the same initial conditions and same equations. By a standard
argument, this implies the convergence of
$(\tilde{y}_{\delta},\tilde{\eta}_{\delta})$ to $(\tilde{y},\tilde{\eta})$. ∎

##  5\. Discussion

###  5.1. The function $\mathfrak{F}$ and the constant $C_{f}$

Recall that the constant $C_{f}$ in the length asymptotics of Theorem C is
given by

$C_{f}=\int_{-\pi/2}^{\pi/2}\mathfrak{F}\left(\frac{1}{\cos(\vartheta)}\right)\,d\vartheta\quad\text{
where }\ \mathfrak{F}(\sigma)=\lim_{\epsilon\to
0}\frac{F^{\prime}(\sigma\epsilon)}{F^{\prime}(\epsilon)}\ \text{ for
}\sigma\geq 1$

with $F$ the inverse function of $f$, see Assumption 10. We now discuss
properties of $\mathfrak{F}$ and $C_{f}$ and their dependency on $f$.

It is useful to introduce a new variable $z$ defined by $x=e^{-z}$, so that
$x\to 0$ corresponds to $z\to\infty$. We do this change of variable in both
the domain and the range of $f$. That is, for given $f$ we consider the
conjugated function

$\phi\coloneqq\overline{\exp}^{-1}\circ
f\circ\overline{\exp}\,,\quad\overline{\exp}(z)=e^{-z}$

or explicitly $\phi(z)=-\log f(e^{-z})$. For the examples in Remark 8 we get,
with $\Phi$ the inverse function of $\phi$:

(38) $\begin{array}[]{c @{\hspace{1cm}} | @{\hspace{1cm}} c @{\hspace{1cm}} |@{\hspace{1cm}} c<EMAIL_ADDRESS>c}f(x)\hfil\hskip 28.45274pt\vrule\hskip 28.45274pt&\phi(z)\hfil\hskip 28.45274pt\vrule\hskip 28.45274pt&\Phi(w)\hfil\hskip 28.45274pt\vrule\hskip 2.84544pt&\text{parameter range}\\\ \hline\cr x^{\alpha}\hfil\hskip 28.45274pt\vrule\hskip 28.45274pt&\alpha z\hfil\hskip 28.45274pt\vrule\hskip 28.45274pt&\frac{1}{\alpha}w\hfil\hskip 28.45274pt\vrule\hskip 2.84544pt&\alpha\geq 1\rule{0.0pt}{11.19443pt}\\\\[5.69054pt] \exp\left(-\ln\left(\frac{1}{x}\right)^{\mu}\right)\hfil\hskip 28.45274pt\vrule\hskip 28.45274pt&z^{\mu}\hfil\hskip 28.45274pt\vrule\hskip 28.45274pt&w^{1/\mu}\hfil\hskip 28.45274pt\vrule\hskip 2.84544pt&\mu>1\\\\[5.69054pt] \exp\left(-\frac{1}{x^{\beta}}\right)\hfil\hskip 28.45274pt\vrule\hskip 28.45274pt&e^{\beta z}\hfil\hskip 28.45274pt\vrule\hskip 28.45274pt&\frac{1}{\beta}\log w\hfil\hskip 28.45274pt\vrule\hskip 2.84544pt&\beta>0\end{array}$

This displays the relative rates of vanishing as $x\to 0$ of $f$ very clearly
since they correspond to rates of growth of $\phi$ as $z\to\infty$: the rate
increases with the parameter in each row, and from top to bottom.

######  Lemma 17.

  1. (a)

Assume $f$ satisfies (2) and $\mathfrak{F}$ exists. Then

(39) $\frac{1}{\sigma}\leq\mathfrak{F}(\sigma)\leq 1$

for all $\sigma\geq 1$ and

$2\leq C_{f}\leq\pi\,.$

  2. (b)

Assume $f$, ${\tilde{f}}$ satisfy (2) and $\mathfrak{F}$,
$\widetilde{\mathfrak{F}}$ exist. Then

(40) $f=O({\tilde{f}})\text{ near zero }\ \Longrightarrow\
\mathfrak{F}\leq\widetilde{\mathfrak{F}}\,,\ \text{ hence }C_{f}\leq
C_{{\tilde{f}}}$

###### Proof.

(a) Since $f$ is increasing and convex, $F$ is increasing and concave, so
$F^{\prime}$ is positive and decreasing. This implies $\mathfrak{F}\leq 1$. By
L’Hôpital’s rule,

(41) $\lim_{\epsilon\to
0}\frac{F(\sigma\epsilon)}{F(\epsilon)}=\sigma\lim_{\epsilon\to
0}\frac{F^{\prime}(\sigma\epsilon)}{F^{\prime}(\epsilon)}=\sigma\mathfrak{F}(\sigma).$

Now $F$ is increasing and $\sigma\geq 1$, so
$\frac{F(\sigma\epsilon)}{F(\epsilon)}\geq 1$, which implies
$\sigma\mathfrak{F}(\sigma)\geq 1$.

Integration yields the inequalities for $C_{f}$.

(b) The inverse functions $F,\Phi$ of $f,\phi$ are related by
$\Phi=\overline{\exp}^{-1}\circ F\circ\overline{\exp}$, or
$F=\overline{\exp}\circ\Phi\circ\overline{\exp}^{-1}$, i.e.
$F(\varepsilon)=\exp(-\Phi(\log\frac{1}{\varepsilon}))$. From this and (41) we
get for each $s\geq 0$

$\lim_{w\to\infty}\left(\Phi(w+s)-\Phi(w)\right)=\log\left(\sigma\mathfrak{F}(\sigma)\right)\,,\quad\sigma=e^{s}\,.$

This means that $\Phi(w)$ has approximate linear growth as $w\to\infty$ at the
rate $\frac{1}{s}\log\left(\sigma\mathfrak{F}(\sigma)\right)$. Now let
$f=O({\tilde{f}})$. Since the existence and value of $\mathfrak{F}$ are not
changed when multiplying $f$ by a constant, we may assume $f\leq{\tilde{f}}$
near zero. This implies $\phi\geq\tilde{\phi}$, so $\Phi\leq\tilde{\Phi}$ near
infinity. Assume for contradiction that
$\mathfrak{F}(\sigma)>\widetilde{\mathfrak{F}}(\sigma)$ for some $\sigma>1$
(they are equal to one for $\sigma=1$). Then $\Phi$ would have a higher linear
growth rate than $\tilde{\Phi}$, so $\Phi(w)>\tilde{\Phi}(w)$ for large $w$,
contradicting $\Phi\leq\tilde{\Phi}$.

∎

For the examples in (38) we now get:

  * •

If $f(x)=x^{\alpha}$ then $\mathfrak{F}(\sigma)=\sigma^{\frac{1}{\alpha}-1}$,
hence $C_{f}=B(1-\tfrac{1}{2\alpha},\tfrac{1}{2})$ where $B$ is the Beta
function.

  * •

In particular, in the conical case $f(x)=x$ we have $\mathfrak{F}(\sigma)=1$,
so $C_{f}=\pi$, and for ‘sharp cusps’, i.e. $f(x)=x^{\alpha}$ with
$\alpha\to\infty$, we get $\mathfrak{F}(\sigma)\to\frac{1}{\sigma}$, so
$C_{f}\to 2$.

  * •

The monotonicity (40) now implies that
$\mathfrak{F}(\sigma)=\frac{1}{\sigma}$, so $C_{f}=2$, for the examples in the
second and third row of (38).

######  Remark 18.

The monotonicity formula (40) shows that $C_{f}=2$ for any worse singularity
than the ones in (38). For instance, define $\exp_{k+1}=\exp\circ\exp_{k}$
with $\exp_{0}(x)=x$. Let $k\geq 0$. Then

$f(x)=\exp\left(-\exp_{k}\left(\frac{1}{x}\right)\right)$

satisfies (2) and by (40), $C_{f}=2$ for any $k\geq 0$.

###  5.2. Example of $\mathfrak{F}$ failing to exist

The idea is to add a fast but suitably small oscillation to $F=f^{-1}$,
preventing convergence. We perturb a family $F(x)$ tending to $0$ as
$x^{\alpha}$ for $\alpha\in(0,1)$.

######  Lemma 19.

Let $\alpha\in(0,1)$ and
$c\geq\frac{2-\alpha}{1-\alpha}\cdot\frac{1+\alpha}{\alpha}$. Define

$F(x)=x^{\alpha}\left(c+\sin(\log(x))\right)=cx^{\alpha}+\tfrac{1}{2i}x^{\alpha+i}-\tfrac{1}{2i}x^{\alpha-i}.$

Here $i=\sqrt{-1}$. Then $F$ is positive, increasing and concave, but
$\lim\limits_{\epsilon\to
0}\frac{F^{\prime}(\epsilon\sigma)}{F^{\prime}(\epsilon)}$ fails to exist
unless $\log(\sigma)=2n\pi$ for some $n\in\mathbb{N}_{0}$.

###### Proof.

We compute

$F^{\prime}(x)=x^{\alpha-1}\left(\alpha c+\alpha\sin\log x+\cos\log x\right),$
$F^{\prime\prime}(x)=x^{\alpha-2}\left(\alpha(\alpha-1)c+(\alpha^{2}-\alpha-1)\sin\log
x+(2\alpha-1)\cos\log x\right).$

$F^{\prime}(x)>0$ clearly holds if $c>\frac{1+\alpha}{\alpha}$, which is true
since $c\geq\frac{2-\alpha}{1-\alpha}\cdot\frac{1+\alpha}{\alpha}$. Similarly,
since $\alpha^{2}-\alpha-1<0$ and $|2\alpha-1|<1$,

$F^{\prime\prime}(x)\leq
x^{\alpha-2}\left(\alpha(\alpha-1)c-(\alpha^{2}-\alpha-1)+1\right)\leq 0$

if $c\geq\frac{2-\alpha}{1-\alpha}\cdot\frac{1+\alpha}{\alpha}$.

To see that the limit fails to exist, assume it did. Then, by L’Hôpital’s
rule, the limit $\lim\limits_{\epsilon\to
0}\frac{F(\epsilon\sigma)}{F(\epsilon)}$ would also exist. But

$\displaystyle\frac{F(\epsilon\sigma)}{F(\epsilon)}=\sigma^{\alpha}\frac{c+\sin\log(\epsilon\sigma)}{c+\sin\log\epsilon}=\sigma^{\alpha}\frac{c+\sin\left(\log\epsilon+\log(\sigma)\right)}{c+\sin\log\epsilon},$

and the existence of the limit as $\epsilon\to 0$ is equivalent to the
existence of the limit

$\lim_{x\to\infty}\frac{c+\sin(x+y)}{c+\sin(x)}$

for all $y=\log(\sigma)\geq 0$. Consider the sequence
$x_{n}=\frac{\pi}{2}+n\pi$. Then

$\frac{c+\sin(x_{n}+y)}{c+\sin(x_{n})}=\frac{c+(-1)^{n}\cos y}{c+(-1)^{n}},$

and this converges only if $\cos y=1$, i.e. $y=\log(\sigma)\in
2\pi\mathbb{N}_{0}$. The converse follows from

$\frac{F^{\prime}(\sigma\epsilon)}{F^{\prime}(\epsilon)}=\sigma^{\alpha-1}$

whenever $\log(\sigma)\in 2\pi\mathbb{N}_{0}$. ∎

We formulate a small question concerning $\mathfrak{F}$.

######  Question 1.

Assume $f$ satisfies (2), and additionally, that

$\lim_{x\to 0}\frac{f(x)}{x^{\alpha}}=0$

for all $\alpha\geq 1$.

Does $\mathfrak{F}$ exist and is $\mathfrak{F}(\sigma)=\frac{1}{\sigma}$?

###  5.3. Why convexity?

One possible extension of our work is to relax the assumptions on $f$. One
could for instance ask what happens when $f$ is concave instead of convex?
What one will notice then is that the bounds on $\mathfrak{F}$ are no longer
true, and $C_{f}$ might very well be infinite. When $f(x)=x^{\alpha}$ with
$0<\alpha<1$, $\mathfrak{F}(\sigma)=\sigma^{\frac{1-\alpha}{\alpha}}$ is
unbounded, and $C_{f}$ is infinite for $\alpha\leq\frac{1}{2}$. The border
case $\alpha=\frac{1}{2}$ can be explicitly solved in the warped case. The
formula (32) holds for arbitrary positive and increasing $f$, hence also for
$f(x)=\sqrt{x}$. Here $F(\rho)=\rho^{2}$, so

$f^{\prime}(\delta)\ell(y_{\delta})=\int_{-\Theta_{\delta}}^{\Theta_{\delta}}\frac{d\theta}{\cos\theta},$

where $\cos\Theta_{\delta}=\sqrt{\frac{\delta}{R}}.$ The integral is
explicitly computable, and the result is

$f^{\prime}(\delta)\ell(y_{\delta})=2\log\left(\frac{R+\sqrt{R^{2}-\delta^{2}}}{\sqrt{\delta}}\right).$

The essential point here is that the length behaves as

$f^{\prime}(\delta)\ell(y_{\delta})\sim\log\delta^{-1},$

so the asymptotic behaviour has become more complicated.

##  6\. Metrics on isolated singularities

Singular spaces typically arise in one of the following ways:

  * •

as subsets of smooth manifolds $M$, e.g. as solution sets $\\{x\in
M:F(x)=0\\}$ of (systems of) equations, where $F:M\to N$ is a smooth map to a
manifold $N$ and singularities may arise at points $p$ where the differential
$dF_{|p}$ is not surjective; an important class of examples are algebraic
varieties;

  * •

as quotients of smooth spaces by non-free group actions; important classes of
examples are orbifolds and moduli spaces.

These spaces are often equipped with natural metrics. For example, complex
projective algebraic varieties carry metrics induced by the Fubini-Study
metric, and the Riemannian moduli space carries a natural metric, which has
(non-isolated) cuspidal singularities.

However, in this section we carefully distinguish non-metric and metric
aspects of singularities. The discussion will be summarized in Subsection 6.3.

###  6.1. Singularities of type $[s]$

We focus on singular spaces arising as subsets of manifolds and only consider
isolated singularities which are described by a profile (or ’shrinking’)
function $s$ of a single variable, in the sense described below. First, we
define:

######  Definition 20.

Let $M$ be a manifold and $X\subset M$. Let $p\in X$. We say that $p$ is an
isolated singularity of $X$ if $X_{0}=X\setminus\\{p\\}$ is a submanifold of
$M$ and $p$ lies in the closure of $X_{0}$.

We define a profile function to be a $C^{1}$ function
$s:[0,\varepsilon)\to[0,\infty)$ for some $\varepsilon>0$ which satisfies

$s(0)=0\,,\ s(z)>0\text{ for }z>0\,.$

We call two profile functions $s,\tilde{s}$ equivalent if $\tilde{s}=as$ near
$0$, where $a$ is positive and $C^{1}$ on some interval $[0,\epsilon)$. Denote
the equivalence class of $s$ by $[s]$.

Let us say that two curves $\gamma$, $\overline{\gamma}$ in $M$ with
$\gamma(0)=\overline{\gamma}(0)=p$ and $\dot{\gamma}(0)\neq
0\neq\dot{\overline{\gamma}}(0)$ are $s$-tangent at $p$ if
$|\gamma(t)-\overline{\gamma}(t)|=O(s(t))$ near $t=0$ in one (hence any) local
coordinate system. If $s(z)=z^{k}$, $k\in\mathbb{N}$, then this corresponds to
tangency of order $k-1$. Roughly, we will say that $X$ has a singularity of
type $s$ at $p$ if it is a union of curves which are pairwise $s$-tangent at
$p$, but not tangent to higher order (i.e. also
$|\gamma(t)-\overline{\gamma}(t)|\geq cs(t)$ for a constant $c>0$).

####  6.1.1. The cuspidal case

We call a profile function $s$ cuspidal if $s^{\prime}(0)=0$. For a cuspidal
profile function $s$ the $s$-blow-down map $\beta_{s}$ is defined by

(42)
$\beta_{s}:[0,\varepsilon)\times\mathbb{R}^{N-1}\to[0,\varepsilon)\times\mathbb{R}^{N-1}\,,\quad(z,u)\mapsto(z,s(z)u)$

It shrinks the hyperplane at height $z$ by the factor $s(z)$. In particular
the boundary plane $z=0$ is collapsed to the origin.

######  Definition 21.

Let $M$ be a manifold and $X\subset M$. Let $p\in X$, and let $s$ be a
cuspidal profile function. We say that $X$ has a cuspidal singularity of type
$[s]$ at $p$ if there are coordinates
$(z,x):U\to\mathbb{R}\times\mathbb{R}^{N-1}$ on a neighbourhood $U\subset M$
of $p$, mapping $p$ to the origin, in terms of which

(43) $X\cap
U=\beta_{s}(\tilde{X})\,,\quad\tilde{X}\coloneqq\bigcup_{z\in[0,\varepsilon)}\\{z\\}\times
Y_{z}$

where $\beta_{s}$ is the $s$-blow-down map and $Y_{z}\subset\mathbb{R}^{N-1}$
are closed submanifolds varying smoothly333That is, $Y_{z}=\iota(Y,z)$ for a
fixed manifold $Y$, with $\iota:Y\times[0,\varepsilon)\to\mathbb{R}^{N-1}$
smooth and $\iota(\cdot,z)$ an embedding for each $z$. Equivalently,
$\tilde{X}$ is a p-submanifold of $\mathbb{R}^{N-1}\times[0,\infty)$, see [12]
or [7]. with $z\in[0,\varepsilon)$. We call $\tilde{X}$ a resolution of $X$.

The curve $z\mapsto(z,0)$ is called axis of the singularity. We also call the
germ of $X\subset M$ at $p$ a singularity of type $[s]$.

These singularities are natural generalizations of surfaces of rotation in
$\mathbb{R}^{3}=\mathbb{R}\times\mathbb{R}^{2}$ with profile function $s$, for
which $Y_{z}=S^{1}\subset\mathbb{R}^{2}$ for each $z$ (see also Example 25).
Compared to isolated singularities of algebraic sets they allow more
flexibility in that $s$ can vanish to infinite order at zero, but are also
more restricted since the blow-down map shrinks each point $u\in Y_{z}$ by the
same factor $s(z)$. For example, the set
$\\{(z,x,y):\,(\frac{x}{z})^{2}+(\frac{y}{z^{2}})^{2}=1\,,\
z>0\\}\cup\\{0\\}\subset\mathbb{R}^{3}$ is not of this type. See [5] for more
on this.

The singularity type is an equivalence class $[s]$ because equivalent profile
functions define the same class of singularities: if $\tilde{s}=as$ then
$s,(Y_{z})$ define the same $X$ as $\tilde{s},(a(z)^{-1}Y_{z})$.

It follows from the cusp condition $s^{\prime}(0)=0$ that a cuspidal
singularity $(X,p)$ has a well-defined tangent direction at $p$, which is a
ray in $T_{p}M$. This is the set of tangents at $p$ of curves in $X$ starting
at $p$, and is equal to the tangent cone in the sense of [1], for instance.

Note that this notion of singularity type is purely differential, i.e. there
is no metric involved. It is also natural in the sense that if the condition
holds in one coordinate system then it holds in any other having the same
axis.444We don’t make use of this. Here is a sketch of the proof: (We assume
here for simplicity that everything is smooth.) We need to show that any
diffeomorphism $\Phi:\mathbb{R}^{N}\to\mathbb{R}^{N}$ which fixes the $z$-axis
pointwise lifts to a diffeomorphism under $\beta_{s}$, locally near zero. By
the argument in [13, Proof of Lemma 2] this reduces to showing that the vector
fields generating these diffeomorphisms lift to smooth vector fields under
$\beta$. These vector fields are spanned over $C^{\infty}$ by the vector
fields $x_{i}\partial_{x_{j}}$ and $x_{i}\partial_{z}$ for $i,j=1,\dots,N-1$.
A simple calculation shows that $\beta_{*}(U_{ij})=x_{i}\partial_{x_{j}}$ for
$U_{ij}=u_{i}\partial_{u_{j}}$ and $\beta_{*}(Z_{i})=x_{i}\partial_{z}$ for
$Z_{i}=su_{i}\partial_{z}-s^{\prime}u_{i}\sum_{j}u_{j}\partial_{u_{j}}$, so
$U_{ij}$, $Z_{i}$ are the desired smooth lifts. A fine point is that the axis
is only determined by $X$ up to a perturbation of order $s$.

####  6.1.2. Conical singularities

Conical singularities can be defined in essentially the same way, where the
profile function is $s(z)=z$ or equivalent to this. However, while cuspidal
singularities always lie in a half space since they have a unique tangent
direction at $p$, it would be unnatural to require this for a cone. Therefore,
we define a conical singularity in the same way as in Definition 21, except
that we replace the map $\beta_{s}$ above by the polar coordinates map

(44)
$\beta_{\mathrm{conic}}:[0,\infty)\times\mathbb{S}^{N-1}\to\mathbb{R}^{N}\,,\quad(z,\omega)\mapsto
z\omega$

and take a smooth family of closed submanifolds
$Y_{z}\subset\mathbb{S}^{N-1}$. The space $[0,\infty)\times\mathbb{S}^{N-1}$
is called the blow-up of $\mathbb{R}^{N}$ at the point $0$ and
$\beta_{\mathrm{conic}}$ the blow-down map, and the definition can be restated
by saying that $X$ is resolved by blowing up $p$. The map $\beta_{s}$ in (42),
with $s(z)=z$, then is $\beta_{\mathrm{conic}}$ written in one of the
projective coordinate systems. See [12] or [7] for details on this.

If $s(z)=z^{k}$ then the map $\beta_{s}$ is the blow-down map for a quasi-
homogeneous blow-up (in projective coordinates; see [9], [11]), so for general
$s$ we have defined a generalized notion of quasi-homogeneous blow-up.

The other remarks above carry over to the conical case. In particular, this
notion of conical singularity is differential, not metric.

###  6.2. Metrics on singularities of type $[s]$

It is natural to consider metrics on $X$ which are induced by smooth metrics
on $M$.

######  Definition 22.

Let $(M,g_{M})$ be a smooth Riemannian manifold and $X\subset M$. Assume that
$X$ has an isolated singularity at $p\in X$. The induced metric for $X$,
denoted $g_{X}$, is the Riemannian metric on $X_{0}=X\setminus\\{p\\}$
obtained by restriction of $g_{M}$ to $TX_{0}$. The induced distance on $X$,
denoted $d_{X}$, is the distance function on $X$ defined by $g_{X}$.

That is, if $q,q^{\prime}\in X$ then $d_{X}(q,q^{\prime})=\inf\ell(\gamma)$
where the infimum is taken over all curves $\gamma:(0,1)\to X_{0}$ with
$\lim_{t\to 0}\gamma(t)=q$, $\lim_{t\to 1}\gamma(t)=q^{\prime}$. This is
sometimes called the intrinsic distance on $X$, and is to be distinguished
from the extrinsic distance, which is the restriction of the distance function
$d_{M}$ (defined on $M\times M$ by $g_{M}$) to $X\times X$.

To understand the geometry of $(X,g)$, e.g. the behaviour of geodesics near
$p$, it is useful to have a normal form of the metric. Specifically, we ask
under which conditions the induced distance makes $X$ a Riemannian space with
conical or cuspidal singularity at $p$ as in Definition 7. This turns out to
be rather subtle even for spaces with cuspidal or conical singularity as
defined above. We discuss this now.

####  6.2.1. Warped products

In order to get a warped product metric on $X$ (rather than just a generalized
one) one expects to have to impose rather rigid conditions. We will consider
the case where $M=\mathbb{R}^{N}$ with the Euclidean metric, and where the
resolution $\tilde{X}$ (defined using standard coordinates $(z,x)$ on
$\mathbb{R}^{N}$) is a product $[0,\varepsilon)\times Y$.

######  Proposition 23.

Let $X\subset\mathbb{R}^{N}$ have an isolated singularity at $0$. Let $g_{X}$
be the metric on $X_{0}$ induced by the Euclidean metric on $\mathbb{R}^{N}$,
and $d_{X}$ the induced distance.

  1. (a)

(Cuspidal case) If $X$ has a cuspidal singularity of type $[s]$, and if, in
(43),

(45) $Y_{z}=Y\ \forall z\ \text{ where }Y\subset\mathbb{R}^{N-1}\text{ is
contained in the unit sphere }\mathbb{S}^{N-2}\text{ centered at }0$

then $g_{X}$ is a warped product metric. The warping function is determined by

(46) $f(r(z))=s(z)\ \text{ where }\
r(z)=\int_{0}^{z}\sqrt{1+(s^{\prime}(w))^{2}}\,dw\,,$

and $h$ is the metric on $Y$ induced by the Euclidean metric on
$\mathbb{R}^{N-1}$. Also, $f(r)\sim s(r)$ as $r\to 0$, and $f$ is convex iff
$s$ is convex, so $(X,d_{X})$ is a Riemannian space with an isolated cuspidal
singularity in this case.

  2. (b)

(Conical case) If $X$ has a conical singularity, and if

(47) $Y_{z}=Y\ \forall z\ \text{ for some $Y\subset\mathbb{S}^{N-1}$}$

then $g_{X}$ is a warped product metric with warping function $f(r)=r$, so
$(X,d_{X})$ is a Riemannian space with an isolated conical singularity. The
metric $h$ is the metric on $Y$ induced by the standard metric on
$\mathbb{S}^{N-1}$.

Note that in the cuspidal case there is an additional condition,
$Y\subset\mathbb{S}^{N-2}$, on the cross section $Y$, while in the conical
case there isn’t. See also the remark after Theorem 27. Condition (47) says
that $X$ is a ‘straight’ cone with an arbitrary base $Y$.

The idea of the proof of part (a) is that $r(z)$ is arc length along the curve
$z\mapsto(z,s(z)u)$ for any fixed $u$ with $|u|=1$. The unit sphere condition
is needed to ensure that mixed terms vanish, i.e. that the curves
$u={\mathrm{const}}.$ are orthogonal to the level sets $z={\mathrm{const}}$.

###### Proof.

  1. (a)

We calculate the induced Riemannian metric on $X_{0}=X\setminus\\{0\\}$.
First, we write the Euclidean metric on $\mathbb{R}^{N}$ in coordinates $z,u$,
i.e. we pull it back under the map $\beta_{s}$, see (42). The differential of
$s(z)u_{i}$ is

$d(s(z)u_{i})=s(z)\,du_{i}+s^{\prime}(z)u_{i}\,dz\,,$

so we get

$\displaystyle\beta_{s}^{*}(g_{\mathrm{eucl}})$
$\displaystyle=dz^{2}+\sum_{i=1}^{N-1}\left(d(su_{i})\right)^{2}$
$\displaystyle=(1+(s^{\prime})^{2}|u|^{2})\,dz^{2}+2ss^{\prime}dz\sum_{i=1}^{N-1}u_{i}\,du_{i}+s^{2}\sum_{i=1}^{N-1}du_{i}^{2}$

where $s=s(z)$ etc. Now we restrict to $Y$. From $Y\subset\mathbb{S}^{N-2}$ we
have that $|u|^{2}$ is constant equal to 1 on $Y$. In particular, the mixed
term, which is $ss^{\prime}\,dz\,d(|u|^{2})$, vanishes when pulled back to
$Y$, so we get

$g_{X_{0}}=(1+(s^{\prime})^{2})\,dz^{2}+s^{2}\,h\,.$

With $r$ defined in (46) we have $dr=\sqrt{1+(s^{\prime})^{2}}\,dz$ and
$s(z)=f(r)$. So in $r,u$ coordinates the metric is $dr^{2}+f(r)^{2}\,h$ as
claimed.

From $s^{\prime}(0)=0$ we get $r(z)\sim z$ as $z\to 0$, hence $f(r)\sim s(r)$
as $r\to 0$. Finally, one calculates

$f^{\prime}(r(z))=\frac{s^{\prime}(z)}{\sqrt{1+(s^{\prime}(z))^{2}}}$

and since the function $z\mapsto r(z)$ as well as the function
$\sigma\mapsto\frac{\sigma}{\sqrt{1+\sigma^{2}}}$ (and hence also their
inverses) are strictly increasing, it follows that $f^{\prime}$ is increasing
iff $s^{\prime}$ is increasing, so $f$ is convex iff $s$ is convex.

  2. (b)

It is well-known that the Euclidean metric on $\mathbb{R}^{N}$ reads
$\beta_{\mathrm{conic}}^{*}(g_{\mathrm{eucl}})=dr^{2}+r^{2}g_{\mathbb{S}^{N-1}}$
in polar coordinates (where we write $r$ instead of $z$ in (44)). Since
$\tilde{X}$ is $\beta^{-1}(X)=[0,R)\times Y$ in polar coordinates, it follows
that $g=dr^{2}+r^{2}h$.

∎

######  Remark 24.

Proposition 23 implies that our results on geodesics apply in the cuspidal
case if (45) holds and $s$ is convex, and in the conical case if (47) holds.
There are also other cases where one obtains conical warped product metrics:

  1. (a)

$X$ is as in Proposition 23(a), except that $s^{\prime}(0)>0$, and $s$ is
convex.
Note in particular the additional condition $Y\subset\mathbb{S}^{N-2}$ in
(45). However, this condition is not needed if $s(z)=cz$ with a constant $c>0$
since then $X$ can be expressed as in Proposition 23(b) (with a different
$Y$).555This can also be seen using a calculation as in the proof of
Proposition 23(a): when we set $r=\sqrt{1+c^{2}|u|^{2}}z$ then $dr^{2}$ will
absorb both the $dz^{2}$ and the mixed term in
$\beta_{s}^{*}(g_{\mathrm{eucl}})$, even if $|u|^{2}$ is not constant on $Y$.
Note that if $s^{\prime}(0)>0$ and $s$ is $C^{2}$ then $f$ is also $C^{2}$, so
by Remark 8 (c) one does not need to require convexity of $s$ then.

  2. (b)

$X$ is as in Proposition 23(a), except that $s(z)=z^{\alpha}$ with
$\alpha\in(0,\frac{2}{3}]$.
(Note that $s$ is not even differentiable at zero then.) The reason is that

$r(z)=\int_{0}^{z}\sqrt{1+\alpha^{2}w^{2\alpha-2}}\,dw=\sum_{k=0}^{\infty}\frac{c_{k}\alpha^{1-2k}}{2k(1-\alpha)+\alpha}z^{2k(1-\alpha)+\alpha}=z^{\alpha}+\beta
z^{2-\alpha}+\dots,$

where $c_{k}$ are the coefficients of the Taylor series of $\sqrt{1+x}$ at
$x=0$, so $\beta=\frac{1}{2\alpha(2-\alpha)}>0$. This implies $f(r)=r-\beta
r^{\gamma}+\dots$ where $\gamma=\frac{2-\alpha}{\alpha}$, which is conical but
not convex. However, if $\alpha\leq\frac{2}{3}$ then it is $C^{2}$ down to
$0$, so $f$ is equivalent to a convex function.
For example, if $s(z)=\sqrt{z}$ and $Y=\mathbb{S}^{N-2}$ then $X$ is in fact
smooth at $p$.

######  Example 25.

If $X\subset\mathbb{R}^{3}$ is a surface of revolution with profile function
$s$ then it has a cuspidal warped product metric if $s$ is cuspidal and a
conical warped product metric if $s$ is $C^{1}$ at zero and $s^{\prime}(0)>0$,
or if $s(z)=z^{\alpha}$ with $\alpha\in(0,\frac{2}{3}]$.

####  6.2.2. Generalized warped products

If we use more general metrics on $M$ near $p$ or do not impose the product
type condition $Y_{z}=Y\,\forall z$ then we cannot expect the induced metric
on $X$ to have warped product structure. We discuss what is known in the
conical and $k$-cuspidal case, i.e. $f(r)=r^{k}$ for some $k\in\mathbb{N}$,
$k\geq 1$.

First, in the conical case one always gets a generalized warped product, as
shown by Melrose and Wunsch.

######  Theorem 26 ([14]).

Let $X\subset M$ have a conical singularity as defined above. Let $g_{M}$ be a
Riemannian metric on $M$ and $d_{X}$ the induced distance function on $X$.

Then $(X,d_{X})$ is a Riemannian space with an isolated conical singularity at
$p$, as in Definition 7.

Theorem 26 is proven in two steps: First one uses normal polar coordinates in
$M$, centred at $p$, to write the induced metric as $g=dr^{2}+r^{2}h$ on
$\tilde{X}=[0,R)\times Y$, where all $Y_{z}$ are identified with a fixed $Y$
and $h$ is a smooth 2-tensor on $\tilde{X}$ restricting to a metric on
$\\{0\\}\times Y$. However, this is more general than (1) since $h$ may
involve $dr^{2}$ terms and also mixed $dr\,dy$ terms. In a second, more
difficult step one proves that one can change coordinates so that these terms
are removed, i.e. $g$ takes the form (1). For this one shows that for each
$q\in Y$ there is a unique geodesic in $(0,R)\times Y$ hitting the boundary at
$(0,q)$, and that these geodesics together define a fibration
$\Phi:[0,R^{\prime})\times Y\to U$ of a neighbourhood $U\subset\tilde{X}$ of
$r=0$. Then $\Phi$ is the desired coordinate system.

The $k$-cuspidal case was analysed by Grandjean and Grieser in [6], with more
detail and a correction given in [2]. Let $X$ be given as in (43) with
$s(z)=z^{k}$, $k\geq 2$, and let $g_{M}$ be any Riemannian metric on $M$.
First, by [6, Proposition 7.3] and [2] a trivialization
$U\cong[0,r_{0})\times{\partial\tilde{X}}$ of a neighbourhood $U$ of
${\partial\tilde{X}}$ can be chosen so that the induced metric on $X$, pulled
back to $\tilde{X}$, has the form

(48) $g=(1+S(z,u)z^{k})\,dz^{2}+z^{2k}h$

for a smooth function $S$ and a smooth 2-tensor $h$ on $\tilde{X}$ restricting
to a metric $h_{0}$ on $\\{0\\}\times\partial\tilde{X}=Y_{0}$.666In [6,
Proposition 7.3] this was stated with $z^{2k-2}$ instead of $z^{k}$, which is
a stronger claim in case $k\geq 3$. However, the proof of that proposition
makes the implicit assumption that the axis of the singularity is tangent to
order $k-1$ to the geodesic in $M$ which starts at $p$ in the same direction
as the axis. This is trivially satisfied if $k=2$ but need not be true if
$k\geq 3$. The proof of the correction and further explanation is given in
[2].

This cannot be improved in general: Consider the subspace $\mathcal{N}\subset
C^{\infty}([0,r_{0})\times{\partial\tilde{X}})$ of functions $f$ satisfying
$d_{\partial\tilde{X}}f=O(z^{k-1})$, and denote by $S_{0}$ the equivalence
class of $S$ modulo $\mathcal{N}$. For example, if $k=2$ then $S_{0}$ can be
identified with the restriction of $S$ to ${\partial\tilde{X}}$ modulo
constants, so it vanishes if and only if $S_{|{\partial\tilde{X}}}$ is
constant. By [6, Lemma 2.2] and [2] the class $S_{0}$ is an invariant of the
singularity and metric. In particular, if $S_{0}$ is non-zero then the metric
can not be put in generalized warped product form by any choice of coordinates
on $\tilde{X}$. However, if $S_{0}$ vanishes then an argument using geodesics
hitting $p$, similar to the argument in [14], shows:

######  Theorem 27 (Theorem 1.2 + Remark 3.4 in [6]).

Suppose $X$ has a $k$-cuspidal singularity as discussed above. If the
equivalence class $S_{0}$ defined above vanishes then the metric on $X$ is a
generalized warped product metric with warping factor $f(r)=r^{k}$.

For example, if $M=\mathbb{R}^{N}$ with the Euclidean metric then
$S_{0}(z,u)=z^{k-2}|u|^{2}$, with $u$ restricted to
$Y_{0}\subset\mathbb{R}^{N-1}$. This shows that the additional condition on
$Y$ in (45) is necessary, up to a constant factor.

In [6] the geodesics hitting ${\partial\tilde{X}}$ were analysed also in the
more general case where $S(z,u)=z^{k-2}\tilde{S}(u)+O(z^{k-1})$ with
$\tilde{S}$ a Morse function on ${\partial\tilde{X}}$.

###  6.3. Summary of Section 6

We distinguish differential and metric notions of conical/cuspidal
singularities.

The differential notion refers to subsets $X$ of a manifold $M$ with isolated
singularity $p\in X$. It is based on the idea of $s$-tangency of curves, where
$s$ is a profile function.

The metric notion is based on the idea of generalized warped product metrics.
It is related to the differential notion as follows. Given a Riemannian metric
$g_{M}$ on $M$, the induced metric $g_{X}$ on $X$ is

  * •

conical if $X$ has a conical singularity in the differential sense

  * •

cuspidal if $X$ has a cuspidal singularity in the differential sense and if it
satisfies additional requirements in its relation to $g_{M}$, like the
vanishing of $S_{0}$ in Theorem 27.

Thus, in the cuspidal case, the metric notion is more restrictive than the
differential notion.

We also mention that there are other notions of conical singularity in the
literature. For example, the notion of corner domain introduced by Dauge in
[4] refers to subsets of $\mathbb{R}^{N}$ which arise from ’straight’ conical
spaces as in Proposition 23(b) by local diffeomorphisms of the ambient space
$\mathbb{R}^{N}$. This is more special than our notion of conical singularity
given in Subsection 6.1.2. (However, corner domains also include non-isolated
singularities such as those arising from the base $Y$ having corner domain
singularities itself.)

##  7\. Acknowledgements

### Funding

The first author was partially supported by DFG Priority Programme 2026
‘Geometry at Infinity’. The second author did not receive any grant for
writing this article.

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