# The Blaschke rolling theorem in Riemannian manifolds of bounded curvature

Kostiantyn Drach  Universitat de Barcelona, Departament de Matemàtiques i
Informàtica, Gran Via 585, 08007 Barcelona, Spain  Centre de Recerca
Mathemàtica<EMAIL_ADDRESS>
###### Abstract.

We generalize the classical Blaschke Rolling Theorem to convex domains in
Riemannian manifolds of bounded sectional curvature and arbitrary dimension.
Our results are sharp and, in this sharp form, are new even in the model
spaces of constant curvature.

###### 2010 Mathematics Subject Classification:

53C40

Acknowledgements. The author is grateful to Alexander Borisenko for many
useful and illuminating discussions on the topic of Blaschke’s theorem
throughout the years, and to Sebastian Boldt for continued encouragement to
finish this project.

##  1\. Introduction

The Blaschke inclusion theorem, a classical result from the field of global
differential geometry, was proven by Wilhelm Blaschke in his famous monograph
(see [4, pp. 114-117]) more than a century ago111The 1st edition of the book
was published in 1916.. He showed the following (see Figure 1):

######  Blaschke’s Inclusion Theorem in $\mathbb{R}^{2}$.

If $B_{1}$ and $B_{2}$ are two strictly convex, compact planar domains with
smooth boundaries touching at some point $p\in\partial B_{1}\cap\partial
B_{2}$ in such a way that they share a common inner normal, then

$B_{1}\subseteq B_{2}$

provided that the curvatures $k_{i}(\nu)$ of the boundaries, as functions of
inner normals $\nu\in\mathbb{S}^{1}$, satisfy

(1.1) $k_{1}(\nu)\geqslant k_{2}(\nu)\text{ for all }\nu\in\mathbb{S}^{1}.$

$p$${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}B_{2}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}B_{1}}$$\nu$
Figure 1. Blaschke’s inclusion theorem.

Informally speaking, if the boundary of $B_{1}$ is “curved more” than the
boundary of $B_{2}$, then $B_{1}$ can “slide” along $\partial B_{2}$ while
staying inside $B_{2}$, as indicated in Figure 1. An interesting special case
is when one of the domains is a disk and thus one of the curvatures $k_{i}$ is
constant. Traditionally, this particular case is distinguished from Blaschke’s
inclusion result and is referred to as the Blaschke rolling theorem — the
corresponding disk can either be rolled inside the domain (along the
boundary), or, vise versa, the domain can roll inside the disk.

Blaschke’s inclusion (and rolling) theorem was later generalized by many
authors. Although extension of the _rolling_ result from curves in
$\mathbb{R}^{2}$ to surfaces in $\mathbb{R}^{n}$, appropriately changing (1.1)
to an inequality between the second fundamental forms, follows from a
section–type (if $B_{2}$ is a ball) or a projection–type (if $B_{1}$ is a
ball) argument, the general inclusion case requires some technical work.
Koutroufiotis [26] generalized Blaschke’s inclusion theorem for complete
curves in $\mathbb{R}^{2}$ and complete surfaces in $\mathbb{R}^{3}$. He also
treated the case when $B_{1}$ and $B_{2}$ touch in a more than one point.
Later, the result was extended to compact surfaces in $\mathbb{R}^{n}$ by
Rauch [29], and to complete surfaces by Delgado [18]. Both of them went along
the original ideas of Blaschke. Brooks and Strantzen [12], by carefully
analyzing previously known arguments, generalized Blaschke’s inclusion theorem
for non-smooth strictly convex sets.

While extensively treated in the Euclidean case, Blaschke’s theorem received
less attention in general Riemannian setting, where similar inclusion-type
questions make perfect sense upon appropriate bounds on curvatures. Karcher
[25] proved the Blaschke rolling theorem for compact domains with the smooth
boundary in model two-dimensional geometries (Euclidean, hyperbolic and
spherical; see also the reference in [25] for some partial results due to
Santaló). Moreover, Karcher showed that, unlike the spherical and Euclidean
cases, Blaschke’s rolling theorem is generally false for convex domains with
an upper bound on the curvature of the boundary lying in the hyperbolic space.
Therefore, one must impose natural restrictions to achieve a proper
generalization (see [25, Section 8]). In [27] Milka, using techniques of
global synthetic geometry, extended the results of Karcher by proving
Blaschke’s rolling theorem in two-dimensional model spaces without
restrictions on smoothness of the boundary by using a concept of specific
curvature of the curve. Milka’s techniques also work for unbounded convex
domains.

For some further progress on the discrete versions of Blaschke’s theorem in
$\mathbb{R}^{2}$ we refer to a paper of Révész [30] and, for convex polygons
in two-dimensional model spaces, to the work of Borisenko and Miquel [11]. The
latter result was partially extended by Borisenko and Miquel to convex
polygons in Hadamard manifolds of dimension $2$ [10].

Multidimensional constant curvature case of Blaschke’s theorem seems to be
folklore in the community due to seemingly straightforward reduction to the
two-dimensional case via projection/section-type arguments or via the so-
called dual map (see, for example, [21, Theorems 9.2.5, 10.4.4] for the
definition and properties of dual maps). The multidimensional Riemannian case
of varying curvature is more delicate. In [23], Howard proved a far reaching
generalization of Blaschke’s rolling theorem for Riemannian manifolds of
bounded geometry with boundary. His generalization concerns the case when a
ball rolls inside a domain and is formulated in terms of equality of rolling
radius $\operatorname{Roll}(M)$ and focal radius
$\operatorname{Focal}(\partial M)$ of a Riemannian manifold $M$ and its
boundary $\partial M$. In particular, [23, Theorems 1.1, 4.4] imply the inner
Blaschke’s rolling theorem for convex domains in constant curvature spaces,
using, for example, [16, Proposition 1.1] to estimate
$\operatorname{Focal}(\partial M)$ in terms of principal curvatures). Using
completely different techniques, the outer Blaschke rolling theorem, that is
when a domain rolls inside a ball, was implicitly proven in $n$-dimensional
hyperbolic space in [9] by Borisenko and Miquel and in the remaining model
spaces of constant curvature, in [7] by Borisenko and the author. The approach
to Blaschke’s theorem in [7] and [9] was based on the so-called radial angle
comparison theorem – a powerful tool to study strictly convex hypersurfaces
with bounds on their principal curvatures, pioneered by Borisenko, and later
extended in the series of publications (e.g., [8, 9, 5, 7]).

In this paper we establish the sharp version of outer Blaschke’s rolling
theorem for uniformly convex domains in Riemannian manifolds of bounded
sectional curvature. As an ingredient, we use a sharp version of the radial
angle comparison theorem, for which we provide a simplified (compared to those
in [9, 7]) proof. In the last section, we will pose some open questions
related to Blaschke’s rolling theorem in spaces of non-constant curvature.

##  2\. Preliminaries and statement of the main results

###  2.1. Preliminaries

Let $(M,\left<\cdot,\cdot\right>)$ be a complete, smooth, orientable
Riemannian manifold of dimension $m\geqslant 2$.

We will consistently use the notation $D_{\Sigma}$ for a closed domain in $M$
with non-empty interior and at least $C^{2}$-smooth boundary $\Sigma:=\partial
D_{\Sigma}$; we assume that $\Sigma$ is connected. Therefore, $\Sigma$ is a
closed embedded orientable hypersurface in $M$. Unless stated otherwise, we do
not assume $D_{\Sigma}$ to be compact.

Denote by $\nu\colon\Sigma\to N\Sigma$ the unit normal vector field along
$\Sigma$ giving a positive orientation to $\Sigma$ with respect to
$D_{\Sigma}$, and for a point $p\in\Sigma$, define $\kappa_{\min}(p)$ to be
the smallest principal curvature of $\Sigma$ at $p$ with respect to $\nu(p)$.

######  Definition 2.1 ($\lambda$-convexity).

Given a constant $\lambda>0$, we say that the hypersurface $\Sigma$ and the
domain $D_{\Sigma}$ are $\lambda$-convex if

$\kappa_{\min}(p)\geqslant\lambda\quad\text{for all}\quad p\in\Sigma.$

Equivalently, if $B^{\nu}$ is the second fundamental form of $\Sigma$ with
respect to $\nu$, then $D_{\Sigma}$ is $\lambda$-convex if and only if

$B^{\nu}(v,v)\geqslant\lambda\left<v,v\right>\quad\text{ for all }\quad
p\in\Sigma,\quad v\in T_{p}\Sigma$

(here, $T_{p}\Sigma$ is the tangent space to $\Sigma$ at $p$). The quantity
$k^{\nu}(p,v):={B^{\nu}|_{p}(v,v)}/{\left<v,v\right>|_{p}}$ is called the
normal curvature of $\Sigma$ at the point $p\in\Sigma$ and in the direction of
$v\in T_{p}\Sigma\operatorname{\setminus}\\{0\\}$. Therefore, a domain is
$\lambda$-convex if and only if all normal curvatures of $\Sigma$ are bounded
below by $\lambda$. Note that in the case $m=2$, that is when $\Sigma$ is a
closed embedded curve, the normal curvature of $\Sigma$ is just the geodesic
curvature of this curve.

The notion of $\lambda$-convexity, also referred to in the literature as
“hyperconvexity”, “spindle convexity”, “ball convexity”, is a generalization
of the classical notion of convexity. It naturally appears in various contexts
in smooth and discrete geometry, most notably, in the recent advances on the
reverse isoperimetric problems (see, e.g., [17, 20] and references therein).

From the definition of $\lambda$-convexity it follows that the second
fundamental form of $\Sigma$ is positive-definite. If $M$ is $C^{2}$-smooth
and $\Sigma$ is $C^{3}$-smooth, then by [3], $D_{\Sigma}$ is a convex domain
in $M$, that is, every two points in $M$ can be joined by a minimizing
geodesic within $D_{\Sigma}$.

######  Remark 2.1.

The notion of $\lambda$-convexity can be extended to non-smooth hypersurfaces
as well; see, e.g., [5, Definition 2.3] and [7]. Although in this paper we
will be focused only on the smooth case, some of the results that we prove can
be easily extended to a non-smooth setting.

Let $M(c)$ a complete simply-connected $m$-dimensional Riemannian manifold of
constant sectional curvature equal $c$; we will refer to such manifolds as to
model spaces of curvature $c$. As usual, we assume $m\geqslant 2$. By a well-
known classification, a model space of curvature $c$ is the spherical space
$\mathbb{S}^{m}(c)$ when $c>0$, the Euclidean space $\mathbb{R}^{m}$ when
$c=0$, and the hyperbolic space $\mathbb{H}^{m}(c)$ when $c<0$. We will
suppress the dimension index when it does not cause confusion.

Recall that a hypersurface $\Sigma$ in a Riemannian manifold is called totally
umbilical (of curvature $\lambda$) if $\kappa^{\nu}(p,v)\equiv\lambda$ for all
$p\in\Sigma$ and $v\in T_{p}\Sigma$. In the model space of curvature $c$ one
can classify complete totally umbilical hypersurfaces of curvature $\lambda$
(see, e.g., [13, Section 35.2.4]). Up to isometry and scaling, for $c\geqslant
0$ these are hyperplanes (that is, complete totally geodesic hypersurfaces)
and geodesic _spheres_. However, in the hyperbolic space there are four types
of totally umbilical hypersurfaces: hyperplanes (when $\lambda=0$),
_equidistants_ (when $1/\sqrt{-c}>\lambda>0$, also called _hyperspheres_),
_horospheres_ (when $\lambda=1/\sqrt{-\lambda}$), and geodesic spheres (when
$\lambda>1/\sqrt{-\lambda}$). Regardless of the value of $c$, spheres are the
only compact totally umbilical hypersurfaces in a model space.

For the later parts, it will be useful to introduce a so-called generalized
sine function

$\operatorname{sn}_{c}(t):=\left\\{\begin{aligned}
&\frac{1}{\sqrt{c}}\sin\left(t\sqrt{c}\right),\text{ if }c>0\\\ &t,\text{ if
}c=0\\\ &\frac{1}{\sqrt{-c}}\sinh\left(t\sqrt{-c}\right),\text{ if }c<0,\\\
\end{aligned}\right.$

and likewise, a generalized cotangent function
$\operatorname{ct}_{c}(t):={\operatorname{sn}_{c}^{\prime}(t)}/{\operatorname{sn}_{c}(t)}$.
In this notation, if $R$ is the radius of a convex geodesic sphere in $M(c)$,
then the normal curvature of this sphere is equal to
$\operatorname{ct}_{c}(R)$. In particular,

(2.1) $\lambda>0,\quad\text{ and moreover }\quad\lambda>\sqrt{-c}\quad\text{
for }\quad c<0.$

We say that $\lambda\in\mathbb{R}$ _satisfies the convex sphere curvature
constraints_ (or _sphere constraints_ for short) if (2.1) holds. For such
$\lambda$, we denote by $R_{\lambda}$ the _characteristic_ radius of the
sphere with curvature $\lambda$ in $M(c)$.

Finally, for a subset $A\subset M$ and a point $x\in M$ the cone over $A$ with
the vertex at $x$, denoted $\mathcal{C}(x,A)$, is the union of all geodesic
segments connecting $x$ to the points in $A$.

We use the notation $\operatorname{inj}(M)$ for the global _injectivity
radius_ of $M$. For a point $x\in M$, the _exponential mapping_
$\exp_{x}\colon T_{x}M\to M$ is well-defined and is a diffeomorphism of the
ball of radius $\operatorname{inj}(M)$ in $T_{p}M$ onto its image. We will say
that a set $S\subset T_{x}M$ _does not belong to any closed half-space_ if for
any hyperplane $\Pi$ passing through the origin in $T_{x}M$, the set $S$ does
not lie in any of the closed half-spaces with respect to $\Pi$.

Furthermore, we denote by $K_{\sigma}=K_{\sigma}(x)$ the sectional curvature
of $M$ along a $2$-dimensional subspace $\sigma\subset T_{x}M$. We will write
$\sec(M)\geqslant c$ if $K_{\sigma}(x)\geqslant c$ for all $x\in M$ and
$\sigma\subset T_{x}M$ (and similarly $\sec(M)\leqslant c$).

###  2.2. Statements of the main results

With the given definitions, we are now ready to state our first main result —
a sharp Riemannian version of Blaschke’s rolling theorem for balls (see Figure
2).

######  Theorem A (Riemannian Blaschke’s ball rolling theorem).

Let $M$ be a complete $C^{2}$-smooth Riemannian manifold of dimension
$m\geqslant 2$ with $\sec(M)\geqslant c$. Suppose $\lambda\in\mathbb{R}$
satisfies the sphere constraints with characteristic radius $R_{\lambda}$, and
assume that $\operatorname{inj}(M)>2R_{\lambda}$. Then every $\lambda$-convex
domain $D_{\Sigma}\subset M$ with $C^{3}$-smooth boundary can roll freely
inside a geodesic ball of radius $R_{\lambda}$ in $M$.

I. More precisely, for every point $s\in\Sigma$, if $\gamma\colon[0,1]\to M$,
$\gamma(0)=s$, $\gamma(1)=\bar{s}$ is the geodesic segment of length
$R_{\lambda}$ such that $\dot{\gamma}(0)$ is the inward pointing unit normal
to $\Sigma$, then the geodesic ball $B(\bar{s},R_{\lambda})$ of radius
$R_{\lambda}$ centered at $\bar{s}$ satisfies

(2.2) $B(\bar{s},R_{\lambda})\supseteq D_{\Sigma}.$

II. Moreover, $\Sigma_{s}:=\Sigma\cap B(\bar{s},R_{\lambda})$ is a (closed)
connected subset of $\Sigma$, and exactly one of the following alternatives
must hold.

  1. (i)

If $\Sigma_{s}$ contains $m+1$ distinct points $s_{1},\ldots,s_{m+1}$ such
that the corresponding $m+1$ vectors $\exp_{\bar{s}}^{-1}(s_{i})$ do not lie
in any closed half-space of $T_{\bar{s}}M$, then $D_{\Sigma}$ is isometric to
a closed ball of radius $R_{\lambda}$ in the model space of curvature $c$.

  2. (ii)

Otherwise, the cone $\mathcal{C}(\bar{s},\Sigma_{x})\subset M$ over
$\Sigma_{s}$ with the vertex at $\bar{s}$ is isometric to a cone in the model
space of curvature $c$ with the vertex at the center of a sphere of radius
$R_{\lambda}$ over a convex domain in this sphere.

Both alternatives (i) and (ii) can occur.

Figure 2. Riemannian Blaschke’s ball rolling theorem.

An immediate corollary to Theorem A is the following diameter comparison
result for $\lambda$-convex domains.

######  Corollary A (Diameter comparison for $\lambda$-convex domains).

If $D\subset M$ be a $\lambda$-convex domain in a Riemannian manifold $M$ with
$\sec(M)\geqslant c$, both satisfying the assumptions of Theorem A, then

$\operatorname{diam}(D)\leqslant
2\operatorname{ct}_{c}^{-1}(\lambda)=\left\\{\begin{aligned}
&\frac{2}{\sqrt{c}}\cot^{-1}\left(\frac{\lambda}{\sqrt{c}}\right),&\quad\text{if
}c>0,\\\ &\frac{2}{\lambda},&\quad\text{if }c=0,\\\
&\frac{2}{\sqrt{-c}}\coth^{-1}\left(\frac{\lambda}{\sqrt{-c}}\right),&\quad\text{if
}c<0&\text{ (and $\lambda>\sqrt{-c}$)}.\\\ \end{aligned}\right.$

In particular, when $c>0$, we have that
$\operatorname{diam}(D)\leqslant\pi/\sqrt{c}$. ∎

Another corollary to Theorem A is the following sharp volume comparison for
$\lambda$-convex domains. It is a generalization to higher dimensions of the
result due to Toponogov [32] (see also Hang–Wang [22, Theorem 4]).

######  Corollary B (Volume comparison for $\lambda$-convex domains).

Let $M$ be a Riemannian manifold with $\sec(M)\geqslant c$ and
$D_{\Sigma}\subset M$ be a $\lambda$-convex domain, both satisfying the
assumptions of Theorem A. Then

$\operatorname{Vol}(D_{\Sigma})\leqslant\operatorname{Vol}(B_{\lambda}),\quad\text{and}\quad\operatorname{Vol}(\Sigma)\leqslant\operatorname{Vol}(\partial
B_{\lambda}),$

where $B_{\lambda}\subset M(c)$ is a ball of radius $R_{\lambda}$ in the model
space of curvature $c$. Moreover, the equality holds if and only if
$D_{\Sigma}$ is isometric to $B_{\lambda}$.

###### Proof.

Inclusion (2.2) applied to $D_{\Sigma}$ implies that
$\operatorname{Vol}(D_{\Sigma})\leqslant\operatorname{Vol}(B(\overline{s},R_{\lambda}))$
and $\operatorname{Vol}(\Sigma)\leqslant\operatorname{Vol}(\partial
B(\overline{s},R_{\lambda}))$ [2] for any $\overline{s}$. By the volume
comparison for geodesic balls [28, Lemma 7.1.3],
$\operatorname{Vol}(B(\overline{s},R_{\lambda}))\leqslant\operatorname{Vol}(B_{\lambda})$
and $\operatorname{Vol}(\partial
B(\overline{s},R_{\lambda}))\leqslant\operatorname{Vol}(\partial
B_{\lambda})$, and the claim follows. The equality case follows by the
rigidity part in Theorem A. ∎

In Theorem A, when $c<0$, we assumed that $\lambda>\sqrt{-c}$ in order to
ensure that in the model hyperbolic space $\mathbb{H}^{m}(c)$ there exists a
ball of curvature $\lambda$. However, as it was mentioned above, in the
hyperbolic space there exist two more types of hypersurfaces that are totally
umbilical, namely, horospheres and equidistants. Below, we state a version of
Riemannian Blaschke’s _horoball_ rolling theorem. We have to restrict to
simply connected negatively curved manifolds in order to define horoballs and
horospheres in the general Riemannian setting.

Recall that a complete simply-connected Riemannian manifold of non-positive
sectional curvature is called a Hadamard manifold. Let $M$ be a Hadamard
manifold, and let $\gamma\colon[0,+\infty)\to M$ be a unit-speed geodesic ray
emanating from $p=\gamma(0)$. (Recall that a unit speed geodesic
$\gamma\colon[0,+\infty)\to M$ is called a geodesics ray if
$\operatorname{d}(\gamma(s),\gamma(t))=|t-s|$ for all $t,s\geqslant$ $\gamma$
and $\lim_{t\to\infty}\gamma(t)$ does not exist, where $\operatorname{d}$ is
the distance in $M$.) Then the equivalence class of unit-speed geodesic rays
$\gamma_{1}$ such that $\operatorname{d}(\gamma(t),\gamma_{1}(t))$ is
uniformly bounded in $t$ defines an ideal boundary point $\gamma(\infty)$ of
$M$. Such rays $\gamma_{1}$ are called asymptotic to $\gamma$.

For a ray $\gamma$ in $M$, we define the Busemann function $b_{\gamma}$ [28,
Section 7.3.2] as

$b_{\gamma}(q):=\lim\limits_{t\to+\infty}(t-\operatorname{d}(q,\gamma(t))).$

In a Hadamard manifold $M$, a ray $\gamma$ emanating from $p=\gamma(0)$
defines a set

$H(\gamma):=\\{q\in M\colon b_{\gamma}(q)\geqslant 0\\},$

which is called a horoball passing through $p$ in the direction of $\gamma$.
The boundary of $H(\gamma)$ is called a horosphere. It contains the point $p$.

Finally, let $M$ be a Hadamard manifold, $\gamma$ be a ray in $M$, and
$A\subset M$ be a set. We define an infinite cone over $A$ with the vertex at
$\gamma(\infty)$, denoted as $\mathcal{C}(\gamma(\infty),A)$, to be the set of
all geodesic rays emanating from a point in $A$ and asymptotic to $\gamma$.

######  Theorem B (Riemannian Blaschke’s horoball rolling theorem).

If $M$ be a $C^{2}$-smooth Hadamard manifold of dimension $m\geqslant 2$ with
$0>\sec(M)\geqslant-\lambda^{2}$, $\lambda>0$, then every $\lambda$-convex
domain $D_{\Sigma}\subset M$ with $C^{3}$-smooth boundary can roll freely in a
horoball in $M$.

More precisely, for every point $s\in\Sigma$ if $\gamma_{s}\colon[0,\infty)\to
M$, $\gamma(0)=s$ is the unique geodesic ray starting at $s$ with
$\dot{\gamma}(0)$ being an inward pointing normal, then

(2.3) $D_{\Sigma}\subseteq H(\gamma_{s}),$

where $H(\gamma_{s})\subset M$ is the horoball passing through $s$ in the
direction of $\gamma_{s}$.

######  Remark 2.2.

Observe that in the statement of Theorem B, $D_{\Sigma}$ may be non-compact.
At the same time, in Theorem A, the domain $D_{\Sigma}$ is necessarily
compact.

As was mentioned in the introduction, the proofs of Theorems A and B are based
on two comparison results, the Radial Angle Comparison theorem from [7], and
Toponogov’s triangle comparison theorem. In section 3, we provide a self-
contained treatment of the former comparison theorem, thus filling some of the
gaps in the existing literature. We pay a particular attention to the equality
case of the Radial Angle Comparison Theorem, which is a novel part. We then
combine both comparison theorems to prove Theorems A and B in Sections 4 are 5
respectively. Finally, in Section 6 we state some open problems related to
Blaschke’s rolling theorems in Riemannian manifolds.

##  3\. The radial angle comparison theorem for $\lambda$-convex domains

In this section, we state and prove the Radial Angle Comparison theorem for
$\lambda$-convex domains in Riemannian manifolds with $\sec\geqslant c$. We
will focus on the case when $\lambda$ satisfies the sphere constraints (2.1).
Some parts of our exposition in this section partly follow [7] and [19].

Let $D_{\Sigma}\subset M$ be a $\lambda$-convex domain. Assume that
$D_{\Sigma}$ is a convex set, i.e., any two points in $D_{\Sigma}$ can be
joint by a geodesic segment within $D_{\Sigma}$. For example, this is true
when $M$ is $C^{2}$-smooth and complete and $\Sigma$ is $C^{3}$-smooth [3],
which we will assume from now on.

Fix a point $p\in D_{\Sigma}$, called _the origin_. Define
$\Sigma_{p}=\Sigma\cap\mathring{B}(p,\operatorname{inj}_{p}(M))$ and
$D_{\Sigma_{p}}=\text{cl}\left(D_{\Sigma}\cap\mathring{B}(p,\operatorname{inj}_{p}(M))\right)$
to be the part of $D_{\Sigma}$ that lies within the open ball of the
injectivity radius of $M$ at $p$ (here $\text{cl}(\cdot)$ is the closure of a
set).

For the origin $p$, we can define the _radial angle function_
$\phi\colon\Sigma_{p}\to[0,\pi)$ as follows. Pick a point $q\in\Sigma_{p}$.
There exists a unique distance minimizing geodesic $\gamma_{pq}\colon[0,1]\to
M$ such that $\gamma_{pq}(0)=p$, $\gamma_{pq}(1)=q$. Then, by definition,
$\phi(q)$ is the smallest angle between $\dot{\gamma}_{pq}(1)$ and $-\nu(p)$:

$\phi(q):=\angle\left(\dot{\gamma}_{pq}(1),-\nu(q)\right),$

where $\nu(q)$ is the inward pointing normal to $\Sigma_{p}$ at $q$. (The
radial angle function depends on the choice of the origin $p$. We will
suppress this dependence where it does not cause a confusion.)

Denote by $\operatorname{dist}\colon M\times M\to\mathbb{R}_{+}$ the distance
function on the manifold $M$. We will also use the notation
$|pq|:=\operatorname{dist}(p,q)$. For the fixed origin point $p\in\Sigma$,
define the restricted distance function
$\operatorname{d}_{p}(\cdot):=\operatorname{dist}(p,\cdot)$ with respect to
$p$. The restricted distance function $\operatorname{d}_{p}$ is $C^{2}$-smooth
in some small neighborhood $\mathcal{U}_{p}$ of $D_{\Sigma_{p}}$ punctured at
$p$.

We can define a smooth restriction
$\operatorname{d}_{p,\Sigma}\colon\Sigma_{p}\to\mathbb{R}_{+}$ by putting

$\operatorname{d}_{p,\Sigma}(q):=\operatorname{d}_{p}(q)=\operatorname{dist}(p,q)\text{
for any }q\in\Sigma_{p}.$

Since $\operatorname{d}_{p}$ is smooth in $\mathcal{U}_{p}$, we can define a
gradient vector field $\operatorname{grad}_{M}\operatorname{d}_{p}$, which is
well-defined smooth section of $T\mathcal{U}_{p}$. Integral trajectories of
this vector field are radial geodesics emanating from $p$. Likewise, we can
define a gradient vector field
$\operatorname{grad}_{\Sigma}\operatorname{d}_{p,\Sigma}$, which is a smooth
section of $T\Sigma_{p}\subset T\mathcal{U}_{p}$.

It is straightforward to see that for each $q\in\Sigma_{p}$ the both gradients
and the radial angle are related by

(3.1)
$\operatorname{grad}_{M}\operatorname{d}_{p}=\operatorname{grad}_{\Sigma}\operatorname{d}_{p,\Sigma}-\cos\phi\cdot\nu\quad\text{in}\quad
T_{q}M.$

This decomposition means that the gradient of the restricted distance function
is just the orthogonal projection of the gradient of the globally defined
distance function onto $T_{q}\Sigma$ viewed as a subspace of $T_{q}M$.

For any $t>0$, denote by $\mathcal{S}(t)$ the geodesic sphere of radius $t$
centered at $p$. Note that $\operatorname{grad}_{M}\operatorname{d}_{p}$ is a
unit normal vector field to $\mathcal{S}(t)\cap\mathcal{U}_{p}$. Since
$\left|\operatorname{grad}_{M}\operatorname{d}_{p}\right|=1$, from (3.1) we
get

$\left|\operatorname{grad}_{\Sigma}\operatorname{d}_{p,\Sigma}\right|=\left|\sin\phi\right|.$

Therefore, $\operatorname{grad}_{\Sigma}\operatorname{d}_{p,\Sigma}$ vanishes
exactly at those points $q$ on $\Sigma_{p}$ where
$\operatorname{grad}_{M}\operatorname{d}_{p}(q)$ and $-\nu(p)$ coincide.
Equivalently, $\operatorname{grad}_{\Sigma}\operatorname{d}_{p,\Sigma}$
vanishes at those $q\in\Sigma_{p}$ where $\Sigma_{p}$ touches
$\mathcal{S}(|pq|)$.

Suppose $\phi(q)\neq 0$. Then in a neighborhood of $q$ in $\Sigma_{p}$ we have
a well-defined unit vector field

$X:=\frac{\operatorname{grad}_{\Sigma}\operatorname{d}_{p,\Sigma}}{\left|\operatorname{grad}_{\Sigma}\operatorname{d}_{p,\Sigma}\right|}.$

Let $\gamma\colon(-\epsilon,\epsilon)\to\Sigma$ be the unique integral
trajectory of $X$ passing through $q$ (see Figure 3):

$\gamma(0)=q,\quad\dot{\gamma}=X.$

The curve $\gamma$ can be uniquely extended in both direction until it either
hits the possibly non-empty boundary of $\Sigma_{p}$, or it hits the point
where $\phi$ vanishes. We will call such a curve the _maximal integral
trajectory_.

$t\\!=\\!\operatorname{d}_{p}(q)$$-\nu$$\operatorname{grad}_{M}\operatorname{d}_{p}$$X$$Y$$\Sigma$$\mathcal{S}(t)$$p$$\gamma$$q$$\phi$
Figure 3. The radial angle and the associated vector fields.

Denote by $s$ the parameter on the maximal integral trajectory $\gamma$ for
$X$. Since $X$ is unit, $s$ is an arc-length parameter. We want to consider
re-parametrization of $\gamma$, namely, suppose

(3.2) $t(s):=\operatorname{d}_{p,\Sigma}(\gamma(s)).$

Since $\operatorname{grad}_{\Sigma}\operatorname{d}_{p,\Sigma}$ doesn’t vanish
along $\gamma$, we can change our parametrization from $s$ to $t$ on the whole
maximal integral trajectory.

Introduce an auxiliary unit vector field $Y$ such that

$\gamma(t)\mapsto
Y(t)\in\operatorname{span}\\{\nu,\operatorname{grad}_{M}\operatorname{d}_{p},X\\}_{\gamma(t)}\subset
T_{\gamma(t)}M,$

$Y(\gamma(t))$ is perpendicular to
$\operatorname{grad}_{M}\operatorname{d}_{p}(\gamma(t))$ and making the angle
$\phi(\gamma(t))$ with $X(\gamma(t))$ (see Figure 3). Therefore, $Y$ is a
well-defined along $\gamma$ vector field that is tangent to $\mathcal{S}(t)$.

The following formula, due to Borisenko, is a multidimensional generalization
of Liouville’s type formula [14, Ch. 4-4, Prop. 4], and, to best of our
knowledge, was first obtained in [8] (see also [1, Lem. 2.2], where the
authors study a similar construction).

######  Proposition 3.1 ([8, 5]).

In the notations above, if $\gamma(t)\subset\Sigma_{p}$ is an integral
trajectory of the vector field $X$ parameterized by the distance parameter $t$
from $p$, then for any $t$ at the point $q=\gamma(t)$ we have the identity

(3.3)
$k^{\nu}\left(q,X\right)=\mu^{-\operatorname{grad}_{M}\operatorname{d}_{p}}\left(q,Y\right)\cdot\cos\phi+\frac{d}{dt}\cos\phi,$

where $\mu^{-\operatorname{grad}_{M}\operatorname{d}_{p}}(q,Y)$ is the normal
curvature of the sphere $\mathcal{S}(t)$ at the point $q$ in the direction
$Y(q)$ with respect to the unit normal
$-\operatorname{grad}_{M}\operatorname{d}_{p}(q)$.

###### Proof.

The proof of this fact that was given in [8] and [5] contains the same
computational inaccuracy. Because of that, we provide a complete proof here.

Suppose $\nabla$ is a Riemannian connection on $M$. We can smoothly extend
both unit vector fields $X$ and $Y$ in some neighborhood of $q$ in $M$; we
will keep the same notation. The result does not depend on the extension. For
brevity, denote $Z:=\operatorname{grad}_{M}\operatorname{d}_{p}$.

From the Gauss decomposition it follows that

(3.4)
$k^{\nu}(q,X)=\left<\nabla_{X}X,\nu\right>,\quad\mu^{-Z}(q,Y)=-\left<\nabla_{Y}Y,Z\right>$

for unit $X$ and $Y$.

We proceeds with a straightforward calculation. By the construction of vector
fields we have the following vector decompositions:

(3.5) $\displaystyle X$ $\displaystyle=\sin\phi\cdot Z+\cos\phi\cdot Y$
$\displaystyle\nu$ $\displaystyle=-\cos\phi\cdot Z+\sin\phi\cdot Y.$

Using (3.4) and (3.5) we compute:

$\displaystyle k^{\nu}\left(q,X\right)$
$\displaystyle=\left<\nabla_{X}X,\nu\right>$
$\displaystyle=\left(\cos\phi\left<Z,\nu\right>-\sin\phi\left<Y,\nu\right>\right)X(\phi)+\sin\phi\left<\nabla_{X}Z,\nu\right>+\cos\phi\left<\nabla_{X}Y,\nu\right>$
$\displaystyle=-X(\phi)+\cos^{2}\phi\left<\nabla_{Y}Y,\nu\right>+\cos\phi\sin\phi\left(\left<\nabla_{Y}Z,\nu\right>+\left<\nabla_{Z}Y,\nu\right>\right)$

(here we used the fact that the integral curves for $Z$ are geodesics, and
thus $\nabla_{Z}Z=0$).

Recall that $Z$ and $Y$ are unit vector fields, and thus
$\left<\nabla_{W}Y,Y\right>=\left<\nabla_{W}Z,Z\right>=0$ for any vector $W\in
T_{q}\Sigma$. Substituting $\nu$ from (3.5) we obtain:

$\displaystyle k^{\nu}\left(q,X\right)$
$\displaystyle=-X(\phi)-\cos^{3}\phi\left<\nabla_{Y}Y,Z\right>$
$\displaystyle+\cos\phi\sin\phi\left(\sin\phi\left<\nabla_{Y}Z,Y\right>-\cos\phi\left<\nabla_{Z}Y,Z\right>\right).$

We can arrange at extension in a neighborhood of $q$ in $M$ of the initial
vector field $Y$ in such a way that $\left<Y,Z\right>=0$ (again, we keep the
same notation for the extension). Then
$\left<\nabla_{Z}Y,Z\right>=\left<Y,\nabla_{Z}Z\right>=0$ and
$\left<\nabla_{Y}Z,Y\right>=-\left<Z,\nabla_{Y}Y\right>$. Altogether we
obtain:

$k^{\nu}\left(q,X\right)=-X(\phi)-\cos\phi\left<\nabla_{Y}Y,Z\right>=X(\phi)+\mu^{-Z}(q,Y)\cdot\cos\phi.$

Finally, observe that

$X(\phi)=X(t)\cdot\frac{d\phi}{dt}=\sin\phi\cdot\frac{d\phi}{dt}=-\frac{d}{dt}\cos\phi$

because of (3.2) and the first decomposition in (3.5). Proposition 3.1 is
proved. ∎

Proposition 3.1 allows us to establish the following comparison-type result
that we call the _Radial Angle Comparison Theorem_ (compare Figure 4).

######  Theorem 3.1 (Radial Angle Comparison Theorem for $\lambda$-convex
domains).

Let $M$ be a $C^{2}$-smooth complete Riemannian manifold of dimension
$m\geqslant 2$ and with

$\sec(K)\geqslant c.$

Suppose that $\lambda\in\mathbb{R}$ satisfies the sphere constraints (2.1),
and assume that $\operatorname{inj}(M)>2R_{\lambda}$ for the radius
$R_{\lambda}$ corresponding to $\lambda$. Let $D_{\Sigma}\subset M$ be a
$\lambda$-convex domain with $C^{3}$-smooth boundary and
$D_{\mathcal{S}_{\lambda}}\subset M(c)$ be a ball of radius $R_{\lambda}$,
with $\mathcal{S}_{\lambda}:=\partial D_{\mathcal{S}_{\lambda}}$. Choose a
point $p\in D_{\Sigma}\operatorname{\setminus}\Sigma$ at distance
$\operatorname{dist}(p,\Sigma)=|ps|<\operatorname{inj}_{p}(M)$ for some
$s\in\Sigma$, and let $\phi\colon\Sigma_{p}\to[0,\pi)$ be the radial angle
function defined on $\Sigma_{p}$ with respect to $p$. Furthermore, let
$p_{\lambda}\in D_{\mathcal{S}_{\lambda}}\setminus\mathcal{S}_{\lambda}$ be a
point such that

$\operatorname{dist}(p,\Sigma)=\operatorname{dist}(p_{\lambda},\mathcal{S}_{\lambda}),$

and let $\phi_{\lambda}\colon\mathcal{S}_{\lambda}\to[0,\pi)$ be the radial
angle function with respect to $p_{\lambda}$.

Then for all pairs of points $q\in\Sigma_{p}$,
$q_{\lambda}\in\mathcal{S}_{\lambda}$ with

$|pq|=|p_{\lambda}q_{\lambda}|,$

for the radial angles satisfy

(3.6) $\phi(q)\leqslant\phi_{\lambda}(q_{\lambda}).$

Moreover, if we have equality in (3.6), then the maximal integral trajectory
$\gamma\colon[0,1)\to\Sigma_{p}$ of the distance function
$\operatorname{d}_{p,\Sigma}\colon\Sigma\to(0,+\infty)$ with $q=\gamma(0)$
terminates at some $s_{1}=\lim_{r\to 1-0}\gamma(r)$ with $|ps_{1}|=|ps|$, and
the cone $\mathcal{C}(p,\gamma[0,1])$ over $\gamma[0,1]=\gamma[0,1)\cup s_{1}$
with the vertex at $p$ is a totally geodesic 2-dimensional submanifold in $M$
isometric to a curvilinear triangle in a disk of radius $R_{\lambda}$ in
$M^{2}(c)$ bounded by two geodesics and an arc of the boundary of the disk.

$M$$M(c)$$\phi(q)$$\phi_{\lambda}(q_{\lambda})$$-\nu$$-\nu_{\lambda}$$q_{\lambda}$$q$$s$$\mathcal{S}_{\lambda}$$\Sigma$$p$$p_{\lambda}$$s_{\lambda}$
Figure 4. The Radial Angle Comparison Theorem.

###### Proof.

Let $s_{\lambda}\in\mathcal{S}_{\lambda}$ be a point such that
$|p_{\lambda}s_{\lambda}|=\operatorname{dist}(p_{\lambda},\mathcal{S}_{\lambda})=|ps|=\operatorname{dist}(p,\Sigma)=:d$.
By construction, $\phi(s)=\phi_{\lambda}(s_{\lambda})=0$.

We want to apply Proposition 3.1. Note that the gradient
$\operatorname{grad}_{M(c)}\operatorname{d}_{p_{\lambda}}$ vanishes on
$\mathcal{S}_{\lambda}$ only at $s_{\lambda}$ and at the point diametrically
opposite to $s_{\lambda}$, that is at the distance $2R_{\lambda}-d$ from
$p_{\lambda}$.

Assume first that $|pq|\neq 2R_{\lambda}-d$. Then
$\operatorname{grad}_{M(c)}\operatorname{d}_{p_{\lambda}}(q_{\lambda})\neq 0$.
And thus if $\operatorname{grad}_{M}\operatorname{d}_{p}(q)=0$, then in (3.6)
we have a strict inequality of the form $0<\phi(q_{\lambda})\neq 0$, which is
true. Therefore, we can assume
$\operatorname{grad}_{M}\operatorname{d}_{p}(q)\neq 0$. We are in position of
applying Proposition 3.1.

Denote $\gamma(t)$ and $\gamma_{\lambda}(t)$ the maximal integral trajectories
of the gradient vector fields given by Proposition 3.1 passing though $q$ and
$q_{\lambda}$ on $\Sigma$ and $\mathcal{S}_{\lambda}$, respectively. Note that
the limit points of $\gamma_{\lambda}$ are precisely $s_{\lambda}$ and its
diametrically opposite point.

By Proposition 3.1, along $\gamma$ and $\gamma_{\lambda}$ we have

(3.7)
$k^{\nu}(t)=\cos\phi(t)\cdot\mu^{\operatorname{grad}_{M}\operatorname{d}_{p}}(t)+\frac{d}{dt}\cos\phi(t),$
(3.8)
$\lambda=\cos\phi_{\lambda}(t)\cdot\mu^{\operatorname{grad}_{M(c)}\operatorname{d}_{p_{\lambda}}}(t)+\frac{d}{dt}\cos\phi_{\lambda}(t),$

where, for brevity, (in the notations from Proposition 3.1)
$k^{\nu}(t)=k^{\nu}(\gamma(t),X)$,
$\mu^{\operatorname{grad}_{M}\operatorname{d}_{p}}(t)=\mu^{\operatorname{grad}_{M}\operatorname{d}_{p}}(\gamma(t),Y)$,
and $\mu^{\operatorname{grad}_{M(c)}\operatorname{d}_{p_{\lambda}}}(t)$ is a
normal curvature of a sphere of radius $t$ in the model space $M(c)$.

Assumptions of the theorem allows us to use the comparison theorem for normal
curvatures geodesic spheres (see [31, Chapter IV, Lemma 2.9]). In our
notations it yields

(3.9)
$\operatorname{ct}_{c}(t)=\mu^{\operatorname{grad}_{M(c)}\operatorname{d}_{p_{\lambda}}}(t)\leqslant\mu^{\operatorname{grad}_{M}\operatorname{d}_{p}}(t).$

Let us subtract from (3.7) equality (3.8); then using (3.9) and the
$\lambda$-convexity condition $k^{\nu}(t)\geqslant\lambda$, we obtain

(3.10) $0\leqslant
k^{\nu}(t)-\lambda\leqslant\left(\cos\phi(t)-\cos\phi_{\lambda}(t)\right)\cdot\operatorname{ct}_{c}(t)+\frac{d}{dt}\left(\cos\phi(t)-\cos\phi_{\lambda}(t)\right).$

If we set $f(t):=\cos\phi(t)-\cos\phi_{\lambda}(t)$, then it follows from
(3.10) that this function satisfies the following differential inequality

(3.11) $f(t)\operatorname{ct}_{c}(t)+\frac{d}{dt}f(t)\geqslant 0.$

Obviously, inequality (3.11) is equivalent to

$\frac{d}{dt}\left(f(t)\operatorname{sn}_{c}(t)\right)\geqslant 0.$

Therefore, the function $f\cdot\operatorname{sn}_{c}$ is monotonically
increasing. Going along $\gamma(t)$ following decreasing $t$ we will
eventually arrive at a limit point, say $s^{*}\in\Sigma$, corresponding to
some distance $t^{*}$ from $p$. Note that $t^{*}\geqslant d$ since $d$ is a
strong minimum of the distance function $\operatorname{d}_{p}(\cdot)$.
Clearly, $\phi(s^{*})=0$ by definition of $s^{*}$. Moreover,
$\gamma_{\lambda}(t^{*})$ is either equal to $s_{\lambda}$ with $t^{*}=d$, or
$\gamma_{\lambda}(t^{*})$ is an interior point of the corresponding maximal
trajectory. In the first case, $f(t^{*})=0$, in the second $f(t^{*})>0$.

In either case, $f(t^{*})\geqslant 0$, and thus monotonicity of
$f\cdot\operatorname{sn}_{c}$ yields (3.6) at $q$ and $q_{\lambda}$, as
claimed.

To finish the proof of inequality (3.6) it is left to consider the case when
$|pq|=2R_{\lambda}-d$. In such case
$\operatorname{grad}_{M(c)}\operatorname{d}_{p_{\lambda}}(q_{\lambda})=0$. If
$\operatorname{grad}_{M}\operatorname{d}_{p}(q)=0$ we are done. If
$\operatorname{grad}_{M}\operatorname{d}_{p}(q)\neq 0$, then there exist a
maximal trajectory $\gamma(t)$ through $p$. Consider a point
$q^{\epsilon}:=\gamma(2R_{\lambda}-d-\epsilon)$ for some small $\epsilon$.
Then we are under conditions of the proven case, and thus obtain
$\varphi(q^{\epsilon})\leqslant\varphi(q^{\epsilon}_{\lambda})$. Sending
$\epsilon\to 0$ we recover $\varphi(q)\leqslant\varphi(q_{\lambda})(=0)$ by
continuity, which finishes the proof of (3.6).

Let us treat the equality case. Assume that
$\phi(q)=\phi_{\lambda}(q_{\lambda})$. Then since
$f(t)\cdot\operatorname{sn}_{c}(t)=(\cos\phi(t)-\cos\phi_{\lambda}(t))\operatorname{sn}_{c}(t)$
with $\operatorname{sn}_{c}(t)\not\equiv 0$ is a monotonically increasing
function, and moreover $f(t^{*})=f(|pq|)=0$, it follows that $f(t)\equiv 0$
along the trajectory $\gamma$ between $q$ and $s^{*}$. But, by construction,
$\phi(t^{*})=0$; hence $\phi_{\lambda}(t^{*})=0$, which is only possible at
$s_{\lambda}$. Therefore $t^{*}=d$. Putting $s_{1}:=s^{*}$, we obtain a
desired point. Let us show that the cone $\mathcal{C}(p,\gamma[d,|pq|])$ is
isometric to a sector of a disk with radius $R_{\lambda}$ in $M^{2}(c)$.

Indeed, since $f(t)\equiv 0$ for all $t\in\left[d,|pq|\right]$, we must have
equality in (3.10) for all such $t$. Taking into account (3.9), this is only
possible when

(3.12)
$\mu^{\operatorname{grad}_{M}\operatorname{d}_{p}}(t)=\operatorname{ct}_{c}(t)$

and

(3.13) $k^{\nu}(t)\equiv\lambda\quad\text{for all }t\in\left[d,|pq|\right].$

For $t\in\left[d,|pq|\right]$, let $\delta_{t}\colon s\mapsto\delta_{t}(s)$,
$s\in[0,t]$, be the unique geodesic segment connecting $p=\delta_{t}(0)$ and
$\gamma(t)=\delta_{t}(t)$ (we assume the arc length parametrization of
$\delta_{t}$). Recall (in the notation of Proposition 3.1) that $Y(t)$ was a
unit vector field defined along $\gamma(t)$ by lying in
$\operatorname{span}\\{\gamma^{\prime}(t),\operatorname{grad}_{M}\operatorname{d}_{p}(\gamma(t))\\}$
and being perpendicular to
$\operatorname{grad}_{M}\operatorname{d}_{p}(\gamma(t))$. Consider the unique
Jacobi field $J(s)\in T_{\delta_{t}(s)}M$ along $\delta_{t}$ such that
$J(0)=0$ and $J(t)=Y(t)$.

By the equality case in the Rauch comparison theorem (see [31, Chapter IV,
Theorem 2.3]), for which the normal curvature comparison (3.9) is a corollary
(see [31, Chapter IV, Lemma 2.9 and Theorem 2.7], we obtain that the sectional
curvature $K_{\sigma}(x)$ of $M$ satisfies

(3.14) $K_{\sigma(s)}(\delta(s))\equiv c,$

where

$\sigma(s):=\operatorname{span}\left\\{d\delta_{t}/ds|_{s},J(s)\right\\}\subset
T_{\delta_{t}(s)}M$

is a 2-dimensional plane spanned by $d\delta_{t}/ds|_{s}$ and $J(s)$.

Put $\tilde{\gamma}(t):=\exp^{-1}_{p}\gamma(t)$, and define
$\delta(t,s):=\exp_{p}(s\tilde{\gamma}(t))$, $(t,s)\in[d,|pq|]\times[0,t]$.
Then $\delta$ defines a smooth 2-dimensional submanifold in $M$, which we
called a cone $\mathcal{C}(p,\gamma)$, such that for a given $t$ the curve
$\delta(t,s)=\delta_{t}(s)$ is a geodesic. The vector field
$I(s):=\partial\delta/\partial t(t,s)$ is a Jacobi field along $\delta_{t}$.
Moreover, $I(0)=0$ and $I(t)$ is proportional to $X(t)$, where $X(t)$ is a
unit vector field tangent to $\gamma(t)$. By construction,
$X(t)\in\operatorname{span}\\{Y(t),\operatorname{grad}_{M}\operatorname{d}_{p}(\gamma(t))\\}$.
Therefore,
$I(t)\in\operatorname{span}\\{J(t),\operatorname{grad}_{M}\operatorname{d}_{p}(\gamma(t))\\}=\operatorname{span}\\{J(t),d\delta_{t}/ds|_{s=t}\\}$.
If $E_{1}(s)=d\delta/ds(s)$, $E_{2}(s),\ldots,E_{m}(s)$ is the orthonormal
frame spanning $T_{\delta_{t}(s)}M$ and parallel along $\delta_{t}(s)$ such
that $\operatorname{span}\\{E_{1}(t),E_{2}(t)\\}=\sigma(t)$, then it follows
that $J(s)\in\operatorname{span}\\{E_{1}(s),E_{2}(s)\\}$ and
$I(s)\in\operatorname{span}\\{E_{1}(s),E_{2}(s)\\}$ for all $s\in[0,t]$. This
implies that
$\sigma(s)=\operatorname{span}\\{d\sigma_{t}/ds|_{s},I(s)\\}=\\{\partial\delta/\partial
s,\partial\delta/\partial t\\}|_{(t,s)}$. Using (3.14) we obtain that
$\mathcal{C}(p,\gamma)$ is a 2-dimensional submanifold of constant curvature
$c$ bounded by two geodesics and a curve of constant geodesic curvature
$\lambda$. The last assertion follows from (3.13). The theorem is proven. ∎

##  4\. Proof of Theorem A

Consider an arbitrary point $s\in\Sigma$, and let $\bar{s}$ and
$B(\bar{s},R_{\lambda})$ be as in the statement of the theorem. Note that the
point $\bar{s}$ with $|s\bar{s}|=R_{\lambda}$ is well defined since
$R_{\lambda}<\operatorname{inj}(M)$. For brevity, put $S:=\partial
B(\bar{s},R_{\lambda})$.

I. Rolling: In order to prove (2.2) it is enough to show

(4.1) $|\bar{s}q|\leqslant R_{\lambda}\quad\text{ for all }\quad
q\in\Sigma\operatorname{\setminus}\\{s\\}.$

For some small $\epsilon<R_{\lambda}$, pick a point $p\in D_{\Sigma}$ such
that $\epsilon=\operatorname{dist}(p,\Sigma)=|pq|$. Such a point always
exists.

We want to apply the Radial Angle Comparison Theorem (Theorem 3.1). For that,
let $\mathcal{S}_{\lambda}$ be a sphere of curvature $\lambda$ in the model
space $M(c)$ centered at $\bar{s}_{\lambda}$, $D_{\mathcal{S}_{\lambda}}$ be
the ball bounded by $\mathcal{S}_{\lambda}$, and consider a point
$p_{\lambda}\in
D_{\mathcal{S}_{\lambda}}\operatorname{\setminus}\mathcal{S}_{\lambda}$ such
that $\operatorname{dist}(p_{\lambda},\mathcal{S}_{\lambda})=\epsilon$ (see
Figure 5). Then for the origins $p$ and $p_{\lambda}$ in $D_{\Sigma}$ and
$D_{\mathcal{S}_{\lambda}}$ we have well-defined radial angle functions $\phi$
and $\phi_{\lambda}$, correspondingly.

Figure 5. Comparison construction in the proof of Theorem A.

There are two possibilities: either $|ps|<2R_{\lambda}-\epsilon$, or
$|ps|\geqslant 2R_{\lambda}-\epsilon$. First assume
$|ps|<2R_{\lambda}-\epsilon$; then there exists a point
$s_{\lambda}\in\mathcal{S}_{\lambda}$ such that

(4.2) $|ps|=|p_{\lambda}s_{\lambda}|.$

By the Radial Angle Comparison Theorem (Theorem 3.1),

(4.3) $\phi(s)\leqslant\phi_{\lambda}(s_{\lambda}).$

Denote by $\Delta$ the geodesic triangle $\triangle\bar{s}sp$ in the
Riemannian manifold $M$, and by $\Delta_{\lambda}$ the geodesic triangle
$\triangle\bar{s}_{\lambda}s_{\lambda}p_{\lambda}$ in $M(c)$. Finally, in the
model space $M(c)$ introduce a comparison triangle
$\tilde{\Delta}=\triangle\tilde{\bar{s}}\tilde{s}\tilde{p}$ such that

$\left|\tilde{\bar{s}}\tilde{s}\right|=\left|\bar{s}s\right|\,(=R_{\lambda}),\quad\left|\tilde{s}\tilde{p}\right|=|sp|,\quad\angle\tilde{\bar{s}}\tilde{s}\tilde{p}=\angle\bar{s}sp\,(=\phi(s)).$

By Toponogov’s triangle comparison theorem (see, e.g., [31, Chapter IV, Thm.
4.2 (2)]) for the triangles $\Delta$ and $\tilde{\Delta}$ we have

(4.4) $|\bar{s}p|\leqslant|\tilde{\bar{s}}\tilde{p}|.$

At the same time, from (4.2), (4.3) and the way we have constructed the point
$\bar{s}_{\lambda}$ it follows that for the elements of the triangles
$\tilde{\Delta}$ and $\Delta_{\lambda}$ in $M(c)$ the relations

$\displaystyle
R_{\lambda}=|\bar{s}_{\lambda}s_{\lambda}|=\left|\tilde{\bar{s}}\tilde{s}\right|,\quad|s_{\lambda}p_{\lambda}|=|\tilde{s}\tilde{p}|,$
$\displaystyle\phi_{\lambda}(s_{\lambda})=\angle\bar{s}_{\lambda}s_{\lambda}p_{\lambda}\geqslant\angle\tilde{\bar{s}}\tilde{s}\tilde{p}=\phi(s)$

hold. Therefore,

(4.5)
$|\bar{s}_{\lambda}p_{\lambda}|\geqslant\left|\tilde{\bar{s}}\tilde{p}\right|.$

Taking into account inequality (4.4) we arrive to the following key inequality

$|\bar{s}p|\leqslant|\bar{s}_{\lambda}p_{\lambda}|.$

By construction,
$|\bar{s}_{\lambda}p_{\lambda}|=R_{\lambda}-\operatorname{dist}(p_{\lambda},\mathcal{S}_{\lambda})=R_{\lambda}-\operatorname{dist}(p,\Sigma)=R_{\lambda}-\epsilon$.
Thus

(4.6) $|\bar{s}p|\leqslant R_{\lambda}-|pq|.$

From the geodesic triangle $\triangle\bar{s}pq$ by the triangle inequality and
inequality (4.6), it follows that

(4.7) $|\bar{s}q|\leqslant|\bar{s}p|+|pq|\leqslant
R_{\lambda}-|pq|+|pq|=R_{\lambda},$

which proves (4.1) under the assumption $|ps|<2R_{\lambda}-\epsilon$.

It remains to consider what will happen if $|ps|\geqslant
2R_{\lambda}-\epsilon$ for any small $\epsilon$. In such situation the radial
angle comparison theorem is not directly applicable. In fact, such situation
is not possible at all, as we will see momentarily.

If $|sq|<2R_{\lambda}$ for any $q\in\Sigma$, then there exits
$\delta_{q}=\delta(q)>0$ such that $|sq|=2R_{\lambda}-\delta_{q}$. Thus if we
choose $\epsilon<\delta_{q}/2$ and $p$ as above ($|pq|=\epsilon$), then

$|sp|\leqslant|sq|+|qp|=2R_{\lambda}-\delta_{q}+|pq|<2R_{\lambda}-\epsilon,$

and we can run the argument above.

Obviously, for $q$ in the neighborhood of $s$ on $\Sigma$ the inequality
$|sq|<2R_{\lambda}$ holds. Let $U(s)\subset\Sigma$ be the maximal such
neighborhood. Then $|sb|=2R_{\lambda}$ for every $b\in\partial U(s)$, if the
boundary is not empty. By the argument above, in particular from (4.1),
$U(s)\subset B(\bar{s},R_{\lambda})$. The geodesic sphere $\mathcal{S}$, as a
subset of $M$, has diameter $2R_{\lambda}$, hence $\partial U(s)$ can be only
a single point diametrically opposite to $s$. Claim (2.2) follows.

II. Equality case: Let us now turn to the equality case. Recall that
$\Sigma_{s}=\Sigma\cap B(\overline{s},R_{\lambda})$. It is clear that
$s\in\Sigma_{s}$; assume that
$\Sigma_{s}\operatorname{\setminus}\\{s\\}\neq\emptyset$. Pick a point
$q\in\Sigma_{s}\operatorname{\setminus}\\{s\\}$, and assume that $q$ is not
diametrically opposite to $s$. Then at $q$ we must have equalities
simultaneously in (4.4), (4.5) and (4.7). Let us analyze each equality
condition.

Equality in (4.4) implies equality case in Toponogov’s comparison theorem (see
[31, Chapter IV, Remark 4.5]) which tells that $\triangle\bar{s}sp$ must span
a totally geodesic 2-dimensional submanifold isometric to a geodesic triangle
in the model space of curvature $c$.

Equality in (4.5) yields equality in the Radial Angle Comparison Theorem
(Theorem 3.1). By decreasing $\epsilon$ we may assume that distance function
$\operatorname{d}_{p}$ has a unique minimum at $q$, and therefore the integral
trajectory $\gamma$ of $\operatorname{d}_{p,\Sigma}$ passing through $s$ will
have $q$ as a limiting point. Applying the equality case in Theorem 3.1 to the
arc $\gamma_{sq}$ of this trajectory between $s$ and $q$ we obtain that the
cone $\mathcal{C}(p,\gamma_{sq})$ must be a 2-dimensional totally geodesic
submanifold of $M$ isometric to a 2-dimensional curvilinear triangle in $M(c)$
bounded by two geodesic segments and an arc of constant geodesic curvature
$\lambda$.

Finally, equality in (4.7) means that we must have equality in a triangle
inequality, which is only possible if the three points $q$, $p$ and $\bar{s}$
lie on the same geodesic.

Combining all the equality conditions above we get the following Claim: if
$q\in\Sigma_{s}\operatorname{\setminus}\\{s\\}$, then the cone
$\mathcal{C}(\bar{s},\gamma_{sq})$ (which, as a set, is equal to
$\Delta\cup\mathcal{C}(p,\gamma_{sq})$) is a totally geodesic 2-dimensional
submanifold of $M$ isometric to a sector in a 2-dimensional disk of radius
$R_{\lambda}$ lying in $M(c)$.

This Claim yields that $|\bar{s}x|=R_{\lambda}$ for every $x\in\gamma_{sq}$.
Therefore, $\gamma_{sq}\subset\Sigma_{s}$ for every
$q\in\Sigma_{s}\operatorname{\setminus}\\{s\\}$ , and thus $\Sigma_{s}$ is
connected.

Let $\bar{s}_{\lambda}$ be a point in $M(c)$, and fix some linear isometry
$I\colon T_{\bar{s}}M\to T_{\bar{s}_{\lambda}}M(c)$ between the tangent spaces
of $M$ and $M(c)$ at $\bar{s}$ and $\bar{s}_{\lambda}$. Define a map

$\Phi:=\exp_{\bar{s}_{\lambda}}\circ I\circ\exp^{-1}_{\bar{s}}\colon M\to
M(c).$

Since $R_{\lambda}<\operatorname{inj}(M)$, the map $\Phi$ sends
$B(\bar{s},R_{\lambda})$ diffeomorphically onto $D_{\mathcal{S}_{\lambda}}$.
Let $\gamma_{s_{\lambda}q_{\lambda}}$ be a unique geodesic segment in
$\mathcal{S}_{\lambda}$ joining $s_{\lambda}=\Phi(s)$ and
$q_{\lambda}=\Phi(q)$. Then, by definition of a cone,
$\mathcal{C}(\bar{s}_{\lambda},\gamma_{s_{\lambda}q_{\lambda}})$ is a
2-dimensional sector in $D_{\mathcal{S}_{\lambda}}$ with the angle less than
$\pi$. By the Claim above, $\Phi$ maps $\mathcal{C}(\bar{s},\gamma_{sq})$
isometrically onto
$\mathcal{C}(\bar{s}_{\lambda},\gamma_{s_{\lambda}q_{\lambda}})$. Hence, the
length of $\gamma_{sq}$ is less than a half of circumference of
$\mathcal{S}_{\lambda}$.

Let $S\subset T_{\bar{s}}M$ be an $m$-dimensional sphere with radius
$R_{\lambda}$ centered at the origin. Define $\operatorname{conv}_{S}(A)$ to
be a convex hull in $S$ of a subset $A\subset S$; we regard $T_{\bar{s}}M$ as
$\mathbb{R}^{m}$ and $S$ as a sphere with the standard metric in
$\mathbb{R}^{m}$. In such terms for a given pair $q_{1},q_{2}\in\Sigma_{s}$ we
showed that

(4.8) $\exp_{\bar{s}}(x)\in\Sigma_{s}\text{ for all
}x\in\operatorname{conv}_{S}\left(\\{\exp_{\bar{s}}^{-1}(q_{1}),\exp_{\bar{s}}^{-1}(q_{2})\\}\right).$

Applying (4.8) inductively, we get that $\Phi(\Sigma_{s})$, as a subset of
$\mathcal{S}_{\lambda}$, must be convex. This yields the equality case under
conditions of case (ii). Otherwise, if we are under the conditions of case (i)
of Theorem A, then

$\operatorname{conv}_{S}(\\{\exp_{\bar{s}}^{-1}(s_{1}),\ldots,\exp_{\bar{s}}^{-1}(s_{m+1})\\})=S,$

which immediately implies that $\Sigma_{s}=\partial B(\bar{s},R_{\lambda})$,
and the conclusion of case (i) follows. The theorem is proven. ∎

##  5\. Proof of Theorem B

The proof of Theorem B is similar to the proof of the inclusion part in
Theorem A.

Let $b^{s}\colon M\to\mathbb{R}$ be a Busemann function associated to the ray
$\gamma_{s}$. Recall that, by definition, $H(\gamma_{s})=\\{x\in M\colon
b^{s}(x)\geqslant 0\\}$ is a closed horoball centered at the ideal boundary
point $\gamma_{s}(\infty)$, while the null set of $b^{s}(\cdot)$ defines the
horosphere $\partial H(\gamma_{s})$. Let $t$ be the arc length parameter for
$\gamma_{s}$ with $\gamma_{s}(0)=s$.

In order to prove (2.3) it is enough to show that

(5.1) $b^{s}(q)\geqslant 0\quad\text{for every}\quad q\in\Sigma.$

Let us consider arbitrary $q\in\Sigma\operatorname{\setminus}\\{s\\}$, and as
in the proof of Theorem A, pick a point $p\in
D_{\Sigma}\operatorname{\setminus}\Sigma$ such that
$\operatorname{dist}(p,\Sigma)=|pq|=\epsilon$ for some small $\epsilon$.

Now we want to define the comparison objects. Let $\mathcal{H}$ be a horoball
in the hyperbolic space $\mathbb{H}(-\lambda^{2})$ defined with respect to a
geodesic ray
$\gamma_{s_{\lambda}}\colon[0,+\infty)\to\mathbb{H}(-\lambda^{2})$,
$\gamma_{s_{\lambda}}(0)=s_{\lambda}$ (we assume that $\gamma_{s_{\lambda}}$
is parameterized by the arc length). The boundary $\partial\mathcal{H}$ (a
horosphere) of $\mathcal{H}$ is a totally umbilical hypersurface of curvature
$\lambda$. Let $b^{s_{\lambda}}(\cdot)$ be the Busemann function corresponding
to $\gamma_{s_{\lambda}}$.

Pick $q_{\lambda}\in\partial\mathcal{H}$ and $p_{\lambda}\in\mathcal{H}$ such
that

$|s_{\lambda}p_{\lambda}|=|sp|,\quad|p_{\lambda}q_{\lambda}|=\operatorname{dist}(p_{\lambda},\partial\mathcal{H})=\epsilon.$

Suppose $\phi$ and $\phi_{\lambda}$ are the radial angle functions defined on
$\Sigma$ and $\partial\mathcal{H}$ with respect to $p$ and $p_{\lambda}$,
respectively. By the Radial Angle Comparison Theorem for $\lambda$-convex
domains in Hadamard manifolds [8, 9], similar to (4.2), we must have

(5.2) $\phi(s)\leqslant\phi_{\lambda}(s_{\lambda}).$

Pick an arbitrary $t>0$. Inequality (5.2) together with Toponogov’s triangle
comparison theorem, analogously to the proof of Theorem A, for the triangles
$\triangle sp\gamma_{s}(t)$ and $\triangle
s_{\lambda}p_{\lambda}\gamma_{s_{\lambda}}(t)$ yields

$|p\gamma_{s}(t)|\leqslant|p_{\lambda}\gamma_{s_{\lambda}}(t)|.$

Therefore, by the triangle inequality,

(5.3)
$|q\gamma_{s}(t)|\leqslant|p\gamma_{s}(t)|+|pq|\leqslant|p_{\lambda}\gamma_{s_{\lambda}}(t)|+\epsilon=|p_{\lambda}\gamma_{s_{\lambda}}(t)|+|p_{\lambda}q_{\lambda}|,$

and thus

(5.4) $t-|q\gamma_{s}(t)|\geqslant
t-|p_{\lambda}\gamma_{s_{\lambda}}(t)|-|p_{\lambda}q_{\lambda}|.$

Sending $t\to\infty$ in the last inequality, we obtain

$b^{s}(q)\geqslant b^{s_{\lambda}}(p_{\lambda})-|p_{\lambda}q_{\lambda}|$

At the same time,

(5.5)
$b^{s_{\lambda}}(p_{\lambda})-|p_{\lambda}q_{\lambda}|=b^{s_{\lambda}}(q_{\lambda})=0$

since $\partial\mathcal{H}$ is horospheres in a hyperbolic space (namely,
since $\partial\mathcal{H}$ is a horosphere and
$q_{\lambda}p_{\lambda}\perp\partial\mathcal{H}$ , the geodesic ray
$q_{\lambda}p_{\lambda}$ is passing through the same ideal boundary point as
$\gamma_{s_{\lambda}}$, yielding the first equality in (5.5); the second
equality in (5.5) follows since $q_{\lambda}\in\partial\mathcal{H}$).
Therefore, $b^{s}(q)\geqslant 0$, which proves (5.1). ∎

##  6\. Open questions

In this section, we present a list of open questions related to the results of
the paper.

Question 1 (Asymptotic rigidity in Toponogov comparison). It is natural to
expect a rigidity part in Theorem B similar to part II of Theorem A. As it
follows from the proof in Section 5, to establish such a rigidity part it
would require proving the following conjectural _asymptotic rigidity_ in
Toponogov’s comparison theorem:

Let $M^{m}$ be an $m$-dimensional Hadamard manifold with
$0>\sec(M^{m})\geqslant-1$. Let $b\colon M^{m}\to\mathbb{R}$ and
$b_{1}\colon\mathbb{H}^{m}(-1)\to\mathbb{R}$ be a pair of Busemann functions
defined with respect to geodesic rays $\gamma\colon[0,+\infty)\to M^{m}$ and
$\gamma_{1}\colon[0,+\infty)\to\mathbb{H}^{m}(-1)$ starting at $x=\gamma(0)$
and $x_{1}=\gamma_{1}(0)$ with the ideal boundary points
$z^{\infty}=\gamma(+\infty)$ and $z_{1}^{\infty}=\gamma_{1}(+\infty)$. Suppose
there exist points $y\in M^{m}$ and $y_{1}\in\mathbb{H}^{m}(-1)$ such that

$|xy|=|x_{1}y_{1}|,\quad
b(y)=b_{1}(y_{1}),\quad\text{and}\quad\angle(\gamma,xy)=\angle(\gamma_{1},x_{1}y_{1}).$

Then the ideal triangle $xyz^{\infty}$ spans a totally geodesic submanifold
isometric to the ideal triangle $x_{1}y_{1}z_{1}^{\infty}$ in
$\mathbb{H}^{2}(-1)$.

Note that in the statement above, $b(x)=b_{1}(x_{1})=0$ by construction.

Question 2 (Equidistant rolling.) For negatively curved Riemannian manifolds,
Theorems A and B establish a Riemannian version of Blaschke’s rolling theorems
in balls and horoballs. Is there a way to give a meaning to the rolling in
convex domains bounded by ‘equidistant hypersurfaces’ in the Riemannian sense?

Question 3 (Inner rolling theorem in Riemannian setting). Dual to the notion
of $\lambda$-convexity, there is a notion of _$\lambda$ -concavity_. A convex
domain $D_{\Sigma}$ is called $\lambda$-concave if the normal curvatures of
$\Sigma$ with respect to the inward pointing normals are bounded _above_ by
$\lambda>0$.

Is there a version of Blaschke’s rolling theorem for $\lambda$-concave domains
in Riemannian manifolds with $\sec\leqslant c$ under the sphere constraints
(2.1)?

In this setting, we expect that the balls roll inside the $\lambda$-concave
domain.

In general, the dual Blaschke’s rolling theorem for the rolling of the ball
inside a domain is not true even for constant curvature spaces, as the
following construction shows. Take a 2-dimensional hyperbolic space in the
unit disk model. Let $o$ be the center of the disk. Consider two disjoint
hyperbolic balls of radii $R_{\lambda}$ symmetric with respect to $o$, and let
$\mathcal{C}$ be the convex hull of these two balls. It is easy to see that
the boundary $\partial\mathcal{C}$ of this hull will have be a convex curve of
geodesic curvature at most $\lambda$ (except 4 points where it is not well
defined). In fact, arcs with curvature $\lambda$ and $0$ will alternate. If
the dual Blaschke’s theorem was true, then a ball of radius $R_{\lambda}$
should roll freely inside $\mathcal{C}$. The last assertion is false, for
instance, for the ball tangent at a closest to $o$ point on
$\partial\mathcal{C}$. This example can be easily modified to make the curve
smooth.

The example above partially motivates the following conjecture: Let $M$ be a
smooth complete simply connected Riemannian manifold of dimension $m\geqslant
2$ with $\sec(M)\leqslant c$. Suppose $\lambda\in\mathbb{R}$ satisfies the
sphere constrains, and $R_{\lambda}$ is the corresponding radius of the
sphere. Let $D_{\Sigma}\subset M$ be a $\lambda$-concave domain with smooth
boundary compactly contained in some ball of radius $\operatorname{inj}(M)/2$.
For $c<0$, additionally assume that $D_{\Sigma}$ is $\sqrt{-c}$-convex. Then a
ball of radius $R_{\lambda}$ can roll freely inside $D_{\Sigma}$.

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