Decoherence and Quantum Measurement:
The Missing Lecture

Stephen D. H<EMAIL_ADDRESS>
Department of Physics and Astronomy
Michigan State University, East Lansing, Michigan 48823, USA

We give an elementary account of quantum measurement and related topics from
the modern perspective of decoherence. The discussion should be comprehensible
to students who have completed a basic course in quantum mechanics with
exposure to concepts such as Hilbert space, density matrices, and von Neumann
projection (“wavefunction collapse”).

##  1 Introduction

Decoherence and quantum measurement are usually neglected in standard courses
on quantum mechanics. While the Shut up and Calculate! perspective is
defensible in an introductory course, students are ultimately entitled to
deeper explanations. We hope these notes are the equivalent of a Missing
Lecture on what has become the widely accepted modern perspective regarding
these topics. The notes are based on material covered in Physics 905 Advanced
Quantum Mechanics, Michigan State University, 2022.

Quantum mechanics, as conventionally formulated, has two types of time
evolution.

1\. An isolated system evolves according to the Schrödinger equation

$i\frac{\partial}{\partial t}\Psi\leavevmode\nobreak\ =\leavevmode\nobreak\
H\Psi\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (1)

The solution satisfies $\Psi(t)=U(t)\Psi(0)\leavevmode\nobreak\ $, and the
time evolution operator $U(t)=\exp(-iHt)$ is unitary: $U^{\dagger}U=1$.

2\. However, when a measurement is made the state undergoes non-unitary von
Neumann projection to an eigenstate corresponding to the observed eigenvalue.

Because the two types of time evolution are so radically different, students
typically demand (and indeed are entitled to demand) a rigorous definition of
exactly when each of them apply. Under what conditions, exactly, does von
Neumann projection occur? When and why does the system deviate from ordinary
Schrödinger evolution?

It is widely acknowledged that the conventional interpretation does not supply
a satisfactory definition for when von Neumann projection (or “wavefunction
collapse”) applies. See, e.g., Against Measurement by J.S. Bell [1].

The modern formulation of quantum mechanics does not require non-unitary
wavefunction collapse. The phenomenology of von Neumann projection can be
reproduced under unitary evolution of the system as a whole, due to a
phenomenon known as decoherence [2, 3, 4, 5], which we explain below. What is
conventionally referred to as measurement is actually a continuous process
that results from entanglement between the measured object and the many
degrees of freedom in the measuring device or local environment.

##  2 Idealized Measurement of a Qubit

Let $Q$ be a single qubit and $M$ a macroscopic device which measures the spin
of the qubit along a particular axis. The eigenstates of spin along this axis
are denoted $|\pm\rangle$. We define the operation of $M$ as follows, where
the combined system is $S=Q+M$. Note $S$ includes both the measuring device
and the qubit $Q$.

$\displaystyle|+\rangle\otimes|M\rangle\longrightarrow|S_{+}\rangle,$ (2)
$\displaystyle|-\rangle\otimes|M\rangle\longrightarrow|S_{-}\rangle,$ (3)

where $S_{+}$ denotes a state of the total system in which the apparatus $M$
has recorded a $+$ outcome (i.e., is in state $M_{+}$), and similarly with
$S_{-}$. See figure (1). We can then ask what happens to a superposition state
$|\Psi_{Q}\rangle=c_{+}|+\rangle+c_{-}|-\rangle$ which enters the device $M$.
In the conventional formulation, with measurement collapse, one of the two
final states $|S_{+}\rangle$ or $\,|S_{-}\rangle$ is realized, with
probabilities $|c_{+}|^{2}$ and $|c_{-}|^{2}$ respectively.

Figure 1: Idealized measurement of a single qubit (spin): $|\pm\rangle$ states
are deflected up or down, entering one of the two detectors. Each detector has
an indicator light which releases a macroscopic number of photons when
activated. In the figure, a $|+\rangle$ spin is deflected upwards and enters
the top detector, causing its indicator light to emit photons. The resulting
state of the system is $|S_{+}\rangle$. A $|-\rangle$ spin state results in
the system state $|S_{-}\rangle$ (not shown), in which the photons are emitted
from the bottom detector. The states $|S_{+}\rangle$ and $|S_{-}\rangle$
differ macroscopically and have almost zero overlap.

However, if the combined system $S=Q+M$ evolves according to the Schrödinger
equation (in particular, linearly), we obtain a superposition of measurement
device states:

$\displaystyle\bigl{(}c_{+}|+\rangle\leavevmode\nobreak\ +\leavevmode\nobreak\
c_{-}|-\rangle\bigr{)}\otimes|M\rangle\leavevmode\nobreak\
\leavevmode\nobreak\ \longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
c_{+}\,|S_{+}\rangle\leavevmode\nobreak\ +\leavevmode\nobreak\
c_{-}\,|S_{-}\rangle.$ (4)

This seems counter to actual experience: measurements produce a single
outcome, not a superposition state.

However, it is almost impossible for an observer in the state $S_{+}$ to be
aware of the second branch of the wave function in state $S_{-}$. Any object
sufficiently complex to be considered either a measuring device or observer
(for example, which can be regarded as semi-classical) will have many degrees
of freedom. A measurement can only be said to have occurred if the states
$M_{+}$ and $M_{-}$ are very different: the outcome of the measurement must be
stored in a redundant and macroscopically accessible way in the device (or,
equivalently, in the local environment). In figure (1), $M_{+}$ corresponds to
the activation of the top detector, and a flash of its light. $M_{-}$
corresponds to the activation of the bottom detector and its light. Typically,
the overlap of $M_{+}$ with $M_{-}$ is effectively zero: of order $\exp(-N)$,
where $N$ is a macroscopic number of degrees of freedom. This tiny overlap
generally persists under further dynamical evolution [6].

For our purposes the phenomenology described above can be taken as a
definition of what we mean by decoherence: two components of a given
superposition state interact with, and become entangled with, environmental or
measuring device degrees of freedom. The resulting environmental components of
the superposition state are radically different and have almost zero overlap.

For All Practical Purposes, to use Bell’s terminology [1], an observer on one
branch can ignore the existence of the other: they are said to have decohered.
Each of the two observers will perceive a collapse to have occurred, although
the evolution of the overall system $S$ has continued to obey the Schrödinger
equation.

We can formalize this analysis using a density matrix. Taking $c_{\pm}$ equal
for simplicity, the post-measurement state is

$|\Psi\rangle=\frac{1}{\sqrt{2}}\,\Bigl{(}|+,M_{+}\rangle\leavevmode\nobreak\
+\leavevmode\nobreak\ |-,M_{-}\rangle\Bigr{)}$ (5)
$\rho=|\Psi\rangle\langle\Psi|\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (6)

We obtain a reduced density matrix $\rho_{Q}$, describing only the qubit
degrees of freedom, by tracing over the $M$ degrees of freedom. See Appendix
for more details.

$\rho_{Q}=\frac{1}{2}\,\Bigl{(}|+\rangle\langle+|\leavevmode\nobreak\
+\leavevmode\nobreak\ |-\rangle\langle-|\leavevmode\nobreak\
+\leavevmode\nobreak\ \langle
M_{-}|M_{+}\rangle\,|+\rangle\langle-|\leavevmode\nobreak\
+\leavevmode\nobreak\ \langle M_{+}|M_{-}\rangle\,|-\rangle\langle+|\Bigr{)}$
(7)

In a properly designed measurement, the states $M_{\pm}$ are macroscopically
distinct, with very small overlap. Therefore, we have

$\rho_{Q}\approx\begin{bmatrix}1/2&0\\\ 0&1/2\end{bmatrix}$ (8)

which describes a mixed state. A mixed state is not a superposition, but
rather can be interpreted as a classical ensemble of outcomes with certain
probabilities. In this case, there are two outcomes (spin up or down), each
with probability $1/2$. Recall, the density matrix describing a pure state
satisfies ${\rm tr}\,\rho^{2}=1$, which no longer holds here.

We see that decoherence can, For All Practical Purposes, reproduce the
phenomenology of von Neumann projection: there are two possible outcomes, and
only one is perceived by a semiclassical observer.

##  3 Macroscopic Superposition and Environmental Decoherence

Decoherence is also responsible for the emergence of a semiclassical reality
from the quantum realm, due to interactions with the environment.

Consider a macroscopic object localized at spatial position $x$. Now form a
superposition state with support at two different locations $x_{1}$ and
$x_{2}$ – i.e., a superposition of the object localized at $x_{1}$ with
another state in which it is localized at $x_{2}$. What happens when this
superposition state interacts with the environment? Assume for example that
the environment contains air molecules and photons which scatter from the
object. The environment near the object is altered by its presence. Let
$E_{1}$ be the environment state which results from interactions with the
object at $x_{1}$ and $E_{2}$ be the environment state after interaction with
the object at $x_{2}$. For example, when the object is at $x_{1}$ the air
molecules near $x_{1}$ scatter from it, but near $x_{2}$ the air molecules are
undisturbed. Placing the object at $x_{2}$ rather than at $x_{1}$ reverses
this pattern, so clearly $E_{1}$ and $E_{2}$, describing the air molecules,
are very different environmental states. See figure (2).

Figure 2: Interactions between the environment (e.g., air molecules) and a
macroscopic object localized at $x_{1}$ lead to a modified environmental state
$|E_{1}\rangle$, with scattered air molecules in the region near $x_{1}$. If
the object is placed at $x_{2}$ the resulting modified environmental state
$|E_{2}\rangle$ describes scattered air molecules in the region near $x_{2}$.
$|E_{1}\rangle$ and $|E_{2}\rangle$ differ macroscopically and have nearly
zero overlap.

Under (linear) Schrödinger evolution, the superposition state evolves as

$\frac{1}{\sqrt{2}}\bigl{(}\,|x_{1}\rangle+|x_{2}\rangle\,\bigr{)}\otimes|E\rangle\leavevmode\nobreak\
\longrightarrow\leavevmode\nobreak\
\frac{1}{\sqrt{2}}\bigl{(}\,|x_{1}E_{1}\rangle+|x_{2}E_{2}\rangle\,\bigr{)}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (9)

We can express the content of (9) as follows. The macroscopic object becomes
entangled, through interaction, with the many degrees of freedom in the
environment. The two branches of the superposition state $\leavevmode\nobreak\
|x_{1}\rangle+|x_{2}\rangle\leavevmode\nobreak\ $ decohere from each other due
to interactions with the environment $E$, specifically because $E$ evolves
into very different states $E_{1,2}$ depending on the location of the object.

The reduced density matrix describing the object position is

$\rho_{x}=\frac{1}{2}\Bigl{(}|x_{1}\rangle\langle x_{1}|\leavevmode\nobreak\
+\leavevmode\nobreak\ |x_{2}\rangle\langle x_{2}|\leavevmode\nobreak\
+\leavevmode\nobreak\ \langle E_{2}|E_{1}\rangle\,|x_{1}\rangle\langle
x_{2}|\leavevmode\nobreak\ +\leavevmode\nobreak\ \langle
E_{1}|E_{2}\rangle\,|x_{2}\rangle\langle x_{1}|\Bigr{)}\leavevmode\nobreak\ .$
(10)

Because $E_{1}$ and $E_{2}$ have very small overlap, we obtain

$\rho_{x}\approx\frac{1}{2}\bigl{(}\,|x_{1}\rangle\langle
x_{1}|\leavevmode\nobreak\ +\leavevmode\nobreak\ |x_{2}\rangle\langle
x_{2}|\,\bigr{)}\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (11)

This is again a mixed state, not a pure state. At this level of approximation,
For All Practical Purposes, an observer either perceives the object at $x_{1}$
or $x_{2}$, but not both.

Macroscopic superposition states of this kind can now be created in the lab:
for example, drum-like mechanical resonators measuring around 10 microns
across have been placed into superpositions of different vibrational modes
[7]. The state of the macroscopic drum membrane, at a given instant in time,
is a superposition with support in different positions, analogous to the setup
analyzed above. To maintain the superposition state (avoid decoherence) the
drum system has to be isolated from environmental interactions.

##  4 Summary and Further Reading

The purpose of these notes is to make students aware of the modern perspective
on quantum measurement, taking into account decoherence. See [2, 3, 4] for
more comprehensive discussions.

Because of decoherence observers inside the system perceive outcomes
consistent with the von Neumann projection postulate even if the system as a
whole never deviates from Schrödinger time evolution. That is, we can simply
assume that the quantum state of the system (observer, measurement device,
environment, etc.) evolves according to the Schrödinger equation at all times.
The projection postulate is not needed for quantum mechanics to describe the
experiences of observers inside the system.

What is conventionally referred to as measurement is actually a continuous
process that results from entanglement between the measured object and the
many degrees of freedom in the measuring device or local environment.

Distinct branches of macroscopic superposition states lose contact with each
other as they become entangled with environmental degrees of freedom. The
branches are said to decohere from each other.

Under Schrödinger evolution, which is unitary, the branches never fully
disappear from the theory [5]. In fact, isolated quantum systems described by
Schrödinger evolution can be shown to spend most of their time in macroscopic
superposition states [6]. Macroscopic superposition states have been realized
in the laboratory [7, 8] – there is no experimental evidence against the
possibility that you might be in a superposition state as you read this
article [5].

The main open problem is to explain the Born rule, which associates
probabilities $|c_{\pm}|^{2}$ with outcomes of measurements on superposition
states as in (4). In traditional formulations of quantum mechanics the Born
rule is introduced ad hoc together with the projection postulate. Under pure
Schrödinger evolution it requires further justification [9, 10].

##  5 Appendix: Density Matrices and Reduced Density Matrices

A density matrix $\rho$ describes the quantum state of a physical system. It
is a generalization of the state vector $\psi$ which represents a pure state.
For a pure state $\rho=|\psi\rangle\langle\psi|$, and satisfies ${\rm
tr}\,\rho^{2}=1$.

However, density matrices can also represent mixed states. One example of a
mixed state arises due to incomplete information: suppose we are told that
$\psi$ is randomly selected from an ensemble with equal probability of being
in state $|+\rangle$ and $|-\rangle$. Then the density matrix is

$\rho=\frac{1}{2}\Bigl{(}|+\rangle\langle+|\leavevmode\nobreak\
+\leavevmode\nobreak\ |-\rangle\langle-|\Bigr{)}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (12)

Note for this mixed state ${\rm tr}\,\rho^{2}=1/2$.

Another situation in which a mixed state arises is in the description of a
degree of freedom $A$ which is entangled with other degrees of freedom $B$.
While the combined state $\Psi_{AB}$ of $A$ and $B$ may be a pure state, we
cannot describe $A$ by itself as such – it is in a mixed state.

The reduced density matrix describing $A$ is obtained by tracing over $B$
(i.e., summing over the basis vectors of the $B$ Hilbert space):

$\rho_{A}={\rm tr}_{B}\,\rho_{AB}={\rm tr}_{B}\leavevmode\nobreak\
|\Psi_{AB}\rangle\langle\Psi_{AB}|\leavevmode\nobreak\ \leavevmode\nobreak\ .$
(13)

Expectations of operators $O_{A}$ acting on the $A$ subspace are given by
$\langle O_{A}\rangle={\rm tr}\,O_{A}\,\rho_{A}$. Under the assumption of
entanglement between $A$ and $B$, $\rho_{A}$ describes a mixed state. For more
details see, e.g., section 2.4.6 of [4].

##  6 Acknowledgements

The author thanks Alexander Adams, Tim Raben, and Lou Lello for useful
comments.

## References

  * [1] J. S. Bell, Against Measurement, Phys. World 3, 33-40 (1990). https://docs.google.com/file/d/0ByYDxaP-OyVjVXAtV2VqQlVNMTgPDF online.
  * [2] D. Giulini, C. Kiefer, E. Joos, J. Kupsch, I. O. Stamatescu, H. D. Zeh, Decoherence and the Appearance of a Classical World in Quantum Theory, Springer (2003).
  * [3] W. H. Zurek, Decoherence, Einselection, and the Quantum Origins of the Classical, Rev. Mod. Phys. 75, 715-775 (2003).
  * [4] M. Schlosshauer, Decoherence and the Quantum to Classical Transition, Springer (2007).
  * [5] One of the earliest treatments of decoherence theory appears in H. Everett III, The Theory of the Universal Wavefunction (PhD thesis), reprinted in B. S. DeWitt and R. N. Graham (eds.), The Many-Worlds Interpretation of Quantum Mechanics, Princeton University Press (1973). Everett uses nonstandard terminology such as quantum correlation in place of entanglement.
  * [6] R. V. Buniy and S. D. H. Hsu, Macroscopic Superposition States in Isolated Quantum Systems, Found. Phys. 51, no.4, 85 (2021) doi:10.1007/s10701-021-00477-2 [arXiv:2011.11661 [quant-ph]].
  * [7] L. Mercier de L’Epinay et al., Quantum Mechanics-Free Subsystem with Mechanical Oscillators, Science 7 May 2021, Vol 372, Issue 6542, p.625 doi/10.1126/science.abf5389
  * [8] K. S. Lee et al., Entanglement between Superconducting Qubits and a Tardigrade, https://arxiv.org/abs/2112.07978https://arxiv.org/abs/2112.07978.
  * [9] S. D. H. Hsu, On the Origin of Probability in Quantum Mechanics, Mod. Phys. Lett. A 27, 1230014 (2012) doi:10.1142/S0217732312300145 [arXiv:1110.0549 [quant-ph]].
  * [10] S. D. H. Hsu, The Measure Problem in No-Collapse (Many Worlds) Quantum Mechanics, Int. J. Mod. Phys. D 26, no.03, 1730008 (2016) doi:10.1142/S0218271817300087 [arXiv:1511.08881 [quant-ph]].


# A note on the vertizontal curvature of fat bundles

Leonardo F. Cavenaghi  Instituto de Matemática, Estatística e Computação
Científica – Unicamp, Rua Sérgio Buarque de Holanda, 651, 13083-859, Campinas,
SP, Brazil<EMAIL_ADDRESS>
###### Abstract.

In [Zil00], W. Ziller poses a compelling question: given a fat principal
$G$-bundle
$(P,\operatorname{\mathsf{g}})\rightarrow(B,\operatorname{\mathsf{h}})$ with
$\dim G=3$, and $\operatorname{\mathsf{g}}$ representing a Riemannian
submersion metric ensuring that the $G$-orbits are totally geodesic, can one
modify $\operatorname{\mathsf{h}}$ to render all vertical curvatures equal to
$1$? In this note, we establish a rigidity result for fat Riemannian
foliations with bounded holonomy and a specific curvature constraint. Our
result addresses Ziller’s question for fat fiber bundles with compact
structure groups, considering connected compact total spaces under a curvature
constraint that holds on various examples, such as locally symmetric spaces.
Additionally, we assume that all vertizontal curvatures coincide at a point.

###### Key words and phrases:

Nonnegative curvatures, Fat bundles, Positive sectional curvature, Sasakian
manifolds

##  1\. Introduction and Preliminaries

Let $\pi\mathrel{\mathop{\mathchar
58\relax}}F\hookrightarrow(M,\operatorname{\mathsf{g}})\rightarrow(B,\operatorname{\mathsf{h}})$
be a Riemannian submersion where $F$ represents the fiber and $B$ is the base.
Let $\mathcal{V}$ denote the vertical bundle of $\pi$, which contains vectors
tangent to the fibers. We term the _horizontal bundle_ its
$\operatorname{\mathsf{g}}$-orthogonal complementary bundle, denoted as
$\mathcal{H}$ and isometric to $(TB,\operatorname{\mathsf{h}})$. We say that
the Riemannian submersion $\pi$ is _fat_ if, for every point $x$ in $M$ and
every nonzero vector $X$ in $\mathcal{H}_{x}$, the following condition holds
for a local horizontal extension $\widetilde{X}$ of $X$:

(1.1) $[\widetilde{X},\mathcal{H}_{x}]^{\mathcal{V}}=\mathcal{V}_{x}.$

The left-hand side in Equation (1.1) is $2A_{X}\mathcal{H}_{x}$ where
$A\mathrel{\mathop{\mathchar
58\relax}}\mathcal{H}_{x}\times\mathcal{H}_{x}\rightarrow\mathcal{V}_{x}$
stands for the O’Neill tensor of the submersion $\pi$.

Initially introduced in [Wei80], fat bundles offer a valuable framework for
exploring the conditions necessary for metrics with positive sectional
curvature on the total space of fiber bundles, particularly those with totally
geodesic fibers. According to Gray [Gra67] or O’Neill [O’N66], in the presence
of totally geodesic fibers, the unreduced sectional curvature of
$\operatorname{\mathsf{g}}$ at any nontrivial plane $X\wedge V$ for
$X\in\mathcal{H},V\in\mathcal{V}$ is given by

$K_{\operatorname{\mathsf{g}}}(X,V)=|A^{*}_{X}V|_{\operatorname{\mathsf{g}}}^{2},$

where $A^{*}_{X}$ is the $\operatorname{\mathsf{g}}$-dual to $A_{X}$. This
underscores the necessity of the fatness condition for a metric with positive
sectional curvature and totally geodesic fibers on $M$. A plane of the form
$\sigma=X\wedge V$ is termed a _vertizontal plane_. In this manuscript, we
adopt the following convention for the Riemannian tensor

$R_{\operatorname{\mathsf{g}}}(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z,\leavevmode\nobreak\
\forall X,Y,Z\in TM.$

Therefore, we define the sectional curvature of a plane $X\wedge Y$ to be

$\mathrm{sec}_{\operatorname{\mathsf{g}}}(X,Y)=R_{\operatorname{\mathsf{g}}}(X,Y,Y,X)/|X|^{2}|Y|^{2}-\langle
X,Y\rangle^{2}$

where
$R_{\operatorname{\mathsf{g}}}(X,Y,Y,X)=\operatorname{\mathsf{g}}\left(R_{\operatorname{\mathsf{g}}}(X,Y)Y,X\right)$.
We sometimes employ the notation $\langle\cdot,\cdot\rangle$ to shortly denote
$\operatorname{\mathsf{g}}(\cdot,\cdot)$.

As highlighted in [Zil00, Proposition 2.5, p.8], for principal fat bundles
with fiber $F=\mathrm{S}^{3}$ or $\mathrm{SO}(3)$, the base manifold $B$ must
have a dimension multiple of $4$. Furthermore, the only known instances of
principal fat bundles with these fibers occur in $3$-Sasakian manifolds. A
manifold $M$ is labeled $3$-Sasakian if it admits an isometric almost free
action by $\mathrm{S}^{3}$ or $\mathrm{SO}(3)$ with totally geodesic orbits of
curvature $1$, satisfying
$R_{\operatorname{\mathsf{g}}}(U^{*},X)Y=-\operatorname{\mathsf{g}}(U^{*},Y)X{+}\operatorname{\mathsf{g}}(X,Y)U^{*}$
for all $U\in\mathfrak{g}$, the Lie algebra of $G$. The superscript $*$
denotes the action vector field derived from $U$. For these metrics, all
vertizontal curvatures are equal to $1$. In W. Ziller’s notes on fat bundles,
it is asked: “Given a fat principal $G$-bundle with $\dim G=3$, is it possible
to change the metric on $B$ such that all vertizontal curvatures are equal to
$1$?” This note shows that this assertion holds for fat fiber bundles with
compact structure groups that are locally symmetric spaces if all vertizontal
curvatures equal $1$ at a point. This follows from the more general rigidity
result, which states for compact connected Riemannian manifolds with fat
Riemannian foliations with bounded holonomy (Theorem 1.6) if at a point the
verzitonal curvatures coincide, it coincide on the whole manifold under an
additional curvature constraint. To make a better sense of this, we introduce
further notation.

The connected components of the fibers of the Riemannian submersion
$\pi\mathrel{\mathop{\mathchar
58\relax}}(M,\operatorname{\mathsf{g}})\rightarrow(B,\operatorname{\mathsf{h}})$
consist of an example of a Riemannian foliation. However, a Riemannian
foliation does not always need to be given in this form. In the following, we
employ the same terminology used in Riemannian submersions to Riemannian
foliations. Namely, tangent vectors to leaves are named vertical. For any
$x\in M$, the vector space collecting vertical vectors is named the vertical
space and denoted by $\mathcal{V}_{x}$. Its
$\operatorname{\mathsf{g}}$-complementary vector space is denoted as
$\mathcal{H}_{x}$ and named the horizontal space. Its elements are named
horizontal vectors. We also maintain the notation
$A_{X}Y=\frac{1}{2}[X,Y]^{\mathcal{V}}$ for the O’Neill tensor, where
$X,Y\in\mathcal{H}_{x}$ and $A^{*}_{X}$ its $\operatorname{\mathsf{g}}$-dual
once $X\in\mathcal{H}_{x}$ is fixed. Since fatness is solely related to a
choice of complementary distribution to the fibers (in the Riemannian
submersion case), we employ the same definition for Riemannian foliations. To
wit,

######  Definition 1.1.

Let $(M,\operatorname{\mathsf{g}},\mathcal{F})$ be a Riemannian manifold with
a Riemannian foliation $\mathcal{F}$. We say that $\mathcal{F}$ is fat if, for
every $x\in M$ and every nonzero vector $X$ in $\mathcal{H}_{x}$, the
following condition holds for a local horizontal extension $\widetilde{X}$ of
$X$:

(1.2) $[\widetilde{X},\mathcal{H}_{x}]^{\mathcal{V}}=\mathcal{V}_{x}.$

######  Definition 1.2 (Holonomy Fields and Dual Holonomy Fields).

Let $(M,\operatorname{\mathsf{g}},\mathcal{F})$ be a compact Riemannian
manifold with a Riemannian foliation $\mathcal{F}$. Let $c$ be a horizontal
curve on $M$. The _holonomy field_ $\xi$ along $c$ is a vertical field
satisfying

(1.3) $\nabla_{\dot{c}}\xi=-A^{*}_{\dot{c}}\xi-S_{\dot{c}}\xi,$

where $S$ is the shape operator, i.e., the $\operatorname{\mathsf{g}}$-dual to
the second fundamental form of the leaves. Likewise, a vertical field $\nu$
along $c$ is called a _dual holonomy field_ if

(1.4) $\nabla_{\dot{c}}\nu=-A^{*}_{\dot{c}}\nu+S_{\dot{c}}\nu.$

For a horizontal curve $c\mathrel{\mathop{\mathchar 58\relax}}[0,1]\to M$, let
$h\mathrel{\mathop{\mathchar
58\relax}}\mathcal{V}_{c(0)}\to\mathcal{V}_{c(1)}$ be the linear isomorphism
given by $h(\xi_{0})=\xi(1)$, where $\xi(t)$ is the holonomy field along $c$
with initial condition $\xi(0)=\xi_{0}$. Following [Spe18], we name $h$ an
infinitesimal holonomy transformation.

######  Definition 1.3 (Bounded Holonomy).

We say that a Riemannian foliation has bounded holonomy if there is a constant
$L$ such that, for every holonomy field $\xi$, $|\xi(1)|\leq L|\xi(0)|$.

######  Example 1.4 (Proposition 3.4 in [Spe18]).

If the structure group of the Riemannian submersion
$\pi\mathrel{\mathop{\mathchar
58\relax}}(M,\operatorname{\mathsf{g}})\rightarrow(B,\operatorname{\mathsf{h}})$
with compact connected total space is compact, the Riemannian foliation
induced by the connected component of the fibers has bounded holonomy.

Let $(M,\operatorname{\mathsf{g}},\mathcal{F})$ be a compact connected
Riemannian manifold with a Riemannian foliation $\mathcal{F}$ with totally
geodesic leaves. A triple $\\{X,V,\mathcal{A}\\}$ is said to be a _good
triple_ if $\exp(tV(s))=\exp(sX(t))$ for all $s,t\in\mathbb{R}$, where
$V(s),X(t)$ denote the Jacobi fields along $\exp(sV)$ and $\exp(tX)$,
respectively, that satisfy $V(0)=V,X(0)=X$ and
$V^{\prime}(0)=\mathcal{A}=X^{\prime}(0)$. Proposition 1.4 in [MT11] ensures
that since the leaves of $\mathcal{F}$ are totally geodesic, then the triple
$\\{X,V,-A^{*}_{X}V\\}$ consists of a good triple for every
$X\in\mathcal{H}_{x}$ and $V\in\mathcal{V}_{x}$. For our purposes, we need the
following:

######  Theorem 1.5 (Munteanu–Tapp, [MT11]).

On any locally symmetric space $(M,\operatorname{\mathsf{g}})$ it holds that
$R_{\operatorname{\mathsf{g}}}(X,V)(A^{*}_{X}V)=0$ where
$R_{\operatorname{\mathsf{g}}}$ is the Riemannian curvature tensor of
$\operatorname{\mathsf{g}}$.

We are in a position to state our results:

######  Theorem 1.6.

Let $(M,\operatorname{\mathsf{g}},\mathcal{F})$ be a compact connected
Riemannian manifold with a fat Riemannian foliation $\mathcal{F}$ with totally
geodesic leaves and bounded holonomy. If for every good triple
$\\{X,V,-A^{*}_{X}V\\}$ we have
$R_{\operatorname{\mathsf{g}}}(X,V)A^{*}_{X}V=0$ then the vertizontal
curvatures of $\operatorname{\mathsf{g}}$ are determined at a point. This
holds, for instance, if $(M,\operatorname{\mathsf{g}})$ is isometric to a
locally symmetric space.

If, in addition, the vertizontal curvatures of $\operatorname{\mathsf{g}}$
coincide at a point, this holds for every point. In particular, a scaling of
$\operatorname{\mathsf{g}}$ exists for which all vertizontal curvatures at
every point equal $1$.

As an immediate corollary we obtain a related result to W. Ziller’s question.

######  Corollary 1.7.

Let $(M,\operatorname{\mathsf{g}})$ be a compact connected Riemannian manifold
with a free isometric action by $G=\mathrm{S}^{3},\leavevmode\nobreak\
\mathrm{SO}(3)$. Assume that the orbits of $G$ in
$(M,\operatorname{\mathsf{g}})$ are totally geodesic and pattern a fat
Riemannian foliation in $(M,\operatorname{\mathsf{g}})$. If

  1. (a)

for all good triples $\\{X,V,-A^{*}_{X}V\\}$ we have
$R_{\operatorname{\mathsf{g}}}(X,V)A^{*}_{X}V=0$

  2. (b)

there exists a point $x\in M$ where all the vertizontal curvatures are equal
to $1$

then $(M,\operatorname{\mathsf{g}})$ is a $3$-Sasakian manifold.

Applying the former to the examples discussed in [Zil00], one recovers the
following by observing that the total spaces below are locally symmetric
spaces with a point with positive vertizontal curvature equal to $1$.

######  Corollary 1.8.

In the following fat $\mathrm{S}^{3},\leavevmode\nobreak\
\mathrm{SO}(3)$-principal bundles a Riemannian submersion metric exists with
constant vertizontal curvature equal to $1$. In particular, the total spaces
are $3$-Sasakian manifolds.

  1. (a)

$\mathrm{S}^{3}\rightarrow\mathrm{S}^{4n+3}\rightarrow\mathbb{H}P^{n}$

  2. (b)

$\mathrm{SO}(3)\rightarrow
T_{1}\mathbb{C}P^{n}\rightarrow\mathrm{Gr}_{2}(\mathbb{C}^{n+1}),\leavevmode\nobreak\
n\geq 2$

  3. (c)

$\mathrm{SO}(3)\rightarrow\mathrm{SO}(n)/\mathrm{S}^{3}\times\mathrm{SO}(n-4)\rightarrow\Lambda_{4,n}^{\circ},\leavevmode\nobreak\
n\geq 5$ (the oriented $4$-planes in $\mathbb{R}^{n}$)

  4. (d)

$\mathrm{S}^{3}\rightarrow\mathrm{Sp}(n+1)/\mathrm{Sp}(n)\rightarrow\mathbb{H}P^{n}$

  5. (e)

$\mathrm{SO}(3)\rightarrow\mathrm{G}_{2}/\mathrm{S}^{3}_{\pm}\rightarrow\mathrm{G}_{2}/\mathrm{SO}(4)$
where $\mathrm{S}^{3}_{\pm}\subset\mathrm{SO}(4)$ are the two simple factors.

##  2\. Proof of Theorem 1.6

Let $(M,\operatorname{\mathsf{g}})$ be a compact connected Riemannian manifold
with a fat Riemannian foliation $\mathcal{F}$ with totally geodesic leaves and
bounded holonomy. Theorem 6.2 in [Spe18] implies that any two points in
$(M,\operatorname{\mathsf{g}})$ can be joined by a smooth horizontal geodesic.

Take any two points $x\neq y\in M$ and let $c\mathrel{\mathop{\mathchar
58\relax}}[0,d_{\operatorname{\mathsf{g}}}(x,y)]\rightarrow M$ to be a smooth
horizontal geodesic joining $x$ to $y$. Varying every possible $y$ once fixed
$x$ (or _vice-versa_) collects every possible horizontal direction. Let $V$ be
a vertical vector at $x$. [MT11, Proposition 1.4] implies that for
$X=\dot{c}(0)$ the set $\\{X,V,-A^{*}_{X}V\\}$ is a good triple, the same
holding for the set $\\{\dot{c}(t),\nu(t),-A^{*}_{\dot{c}(t)}\nu(t)\\}$ for
every $t\in[0,d_{\operatorname{\mathsf{g}}}(x,y)]$ where $\nu(t)$ as a dual-
holonomy field extending $V$ along $c$.

Following [O’N66, Theorem 2] we have

$\displaystyle
R_{\operatorname{\mathsf{g}}}(\dot{c},A^{*}_{\dot{c}}\nu,\dot{c},\nu)$
$\displaystyle=-\langle(\nabla_{\dot{c}}^{\mathcal{V}}A)_{\dot{c}}A^{*}_{\dot{c}}\nu,\nu\rangle$
$\displaystyle=-\langle\nabla_{\dot{c}}(A_{\dot{c}}A^{*}_{\dot{c}}\nu)-A_{\dot{c}}(\nabla_{\dot{c}}A^{*}_{\dot{c}}\nu),\nu\rangle$
$\displaystyle=-\langle\nabla_{\dot{c}}(A_{\dot{c}}A^{*}_{\dot{c}}\nu),\nu\rangle+\langle\nabla_{\dot{c}}A^{*}_{\dot{c}}\nu,A^{*}_{\dot{c}}\nu\rangle$
$\displaystyle=-\frac{d}{dt}\langle
A_{\dot{c}}A^{*}_{\dot{c}}\nu,\nu\rangle+\langle
A^{*}_{\dot{c}}\nu,A^{*}_{\dot{c}}(\nabla_{\dot{c}}\nu)\rangle+\langle\nabla_{\dot{c}}A^{*}_{\dot{c}}\nu,A^{*}_{\dot{c}}\nu\rangle$
$\displaystyle=-\frac{d}{dt}|A^{*}_{\dot{c}}\nu|^{2}+\langle\nabla_{\dot{c}}A^{*}_{\dot{c}}\nu,A^{*}_{\dot{c}}\nu\rangle.$

According to our hypotheses we have
$R_{\operatorname{\mathsf{g}}}(\dot{c},A^{*}_{\dot{c}}\nu,\nu,\dot{c})=0$.
Thus, the former sequence of equalities ensures that

$\frac{d}{dt}|A^{*}_{\dot{c}}\nu|^{2}=\langle\nabla_{\dot{c}}A^{*}_{\dot{c}}\nu,A^{*}_{\dot{c}}\nu\rangle.$

However,

$\displaystyle\frac{d}{dt}K_{\operatorname{\mathsf{g}}}(\dot{c},\nu)=\frac{d}{dt}|A^{*}_{\dot{c}}\nu|_{\operatorname{\mathsf{g}}}^{2}$
$\displaystyle=2\langle
A^{*}_{\dot{c}}\nu,\nabla_{\dot{c}}(A^{*}_{\dot{c}}\nu)\rangle.$

Therefore, $\frac{d}{dt}K_{\operatorname{\mathsf{g}}}(\dot{c},\nu)=0$, this
proves the first assertion.

To the last assertion of the theorem, observe that if there is a point $x\in
M$ where all vertizontal curvature coincides, the former computation ensures
that for any other $y$, these curvatures coincide with that at $x$. More
precisely, assuming that at $x$ we have $K_{\operatorname{\mathsf{g}}}(X,V)=1$
for every non-degenerate plane $X\wedge V$, the set
$\mathcal{M}\mathrel{\mathop{\mathchar 58\relax}}=\\{x\in
M\mathrel{\mathop{\mathchar
58\relax}}K_{\operatorname{\mathsf{g}}}(X,V)=1\leavevmode\nobreak\ \forall
X\in\mathcal{H}_{x},\leavevmode\nobreak\
V\in\mathcal{V}_{x},\leavevmode\nobreak\ X\wedge V\neq 0,\leavevmode\nobreak\
\text{such that}\leavevmode\nobreak\
K_{\operatorname{\mathsf{g}}}(X,V)=1\\}=M$. Indeed, we show that $\mathcal{M}$
is open and closed. Since it is non-empty the connectedness of $M$ will ensure
the result.

Since horizontal and vertical distributions globally form sub-bundles of $TM$,
it is obvious from the continuity of $K_{\operatorname{\mathsf{g}}}$ in the
two-plane Grasmannian that $\mathcal{M}$ is closed. It is left to prove it is
open. However, this already follows from the former step. Due to Theorem 6.2
in [Spe18] one has that a geodesic ball centered at $x$ contains a ball
$\mathcal{B}$ in $M$ where two points can be joined by a horizontal smooth
geodesic. Thus, according the first part of the proof, for any
$X\in\mathcal{H}_{x}$ and any $V\in\mathcal{V}_{x}$, the curvature
$K_{\operatorname{\mathsf{g}}}(\dot{c},\nu)$ is constant along the horizontal
geodesic generated by $X$ where $\nu$ is a dual-holonomy field extension of
$V$ along $c$. Therefore, $\mathcal{M}$ contains $\mathcal{B}$. Thus,
$\mathcal{M}$ is open, as desired.

$\blacksquare$

##  3\. Proof of Corollary 1.8

We recall that a homogeneous bundle is of the form

$K/H\rightarrow G/H\rightarrow G/K$

where $H<K<G$. If $H$ is normal in $K$, then the former homogeneous bundle is
said to be _principal homogeneous_.

Once chosen an $\mathrm{Ad}(K)$-invariant splitting
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{m}$, there exists an invariant
connection $\theta$ on $G\rightarrow G/K$ that induces a horizontal
homogeneous distribution $\widetilde{\mathcal{H}}$ in $G/H\rightarrow G/K$
satisfying $\widetilde{\mathcal{H}}_{(H)}\cong\mathfrak{m}$. Moreover, we can
induce homogeneous metrics on $K/H$ and $G/K$ via the identifications
$T_{(H)}K/H\cong\mathfrak{p},\leavevmode\nobreak\ T_{(K)}G/K\cong\mathfrak{m}$
after choosing an $\mathrm{Ad}(H)$-invariant splitting
$\mathfrak{k}=\mathfrak{h}\oplus\mathfrak{p}$ and
$\mathrm{Ad}(H),\leavevmode\nobreak\ \mathrm{Ad}(K)$-invariant metrics on
$\mathfrak{p}$ and $\mathfrak{m}$, respectively. Gathering this all, we can
define a mertic $\operatorname{\mathsf{g}}$ on $G/H$ satisfying
$T_{(H)}G/H\cong\mathfrak{p}\oplus\mathfrak{m}$. As Proposition 3.1 in [Zil00]
ensures, we have
$K_{\operatorname{\mathsf{g}}}(X,V)=|[X,V]|_{\operatorname{\mathsf{g}}}^{2}$
for $X\in\mathfrak{m},\leavevmode\nobreak\ V\in\mathfrak{p}$.

In the case $K/H=\mathrm{S}^{3},\leavevmode\nobreak\ \mathrm{SO}(3)$, the
former homogeneous fibrations can be described in terms of homogeneous
quaternionic symmetric spaces $G/K.\mathrm{Sp}(1)$ (named Wolf spaces).
Proposition 3.4 in [Zil00] states that if $G/K.\mathrm{Sp}(1)$ is a
quaternionic symmetric space, then the principal bundle fibration
$K.\mathrm{Sp}(1)/K\rightarrow G/K\rightarrow G/K.\mathrm{Sp}(1)$ is fat.
Furthermore, that $K.\mathrm{Sp}(1)/K=\mathrm{S}^{3}$ or $\mathrm{SO}(3)$ and
$\mathrm{Sp}(1)/K$ is 3-Sasakian. Every listed example in the statement of
Corollary 1.8 is of this form. It is also straightforward to check that the
corresponding total spaces are realized as symmetric spaces; see pp.18 in
[Zil00].

Let $\mathbb{H}$ stands to the quaternions and consider the Hopf fibration
$\mathrm{Sp}(1)\rightarrow\mathrm{S}^{4n+1}\rightarrow\mathbb{H}^{n}$.
Adapting the metric construction above to this fibration one trivially has
that the vertizontal curvatres for this fibration are equal to $1$. This fact
concludes the proof since [Zil00, pp. 17] ensures that the restriction of the
isotropy representation of $K.\mathrm{Sp}(1)$ on $G/K.\mathrm{Sp}(1)$ to the
subgroup $\mathrm{Sp}(1)$ is equivalent to the usual Hopf action of
$\mathrm{Sp}(1)$ on $\mathbb{H}^{n}$. Thus, for the metric
$\operatorname{\mathsf{g}}$ constructed as above we have that the vertizontal
curvatures coincide with that of the quaternionic Hopf fibrations.
$\blacksquare$

## Ackowledgements

L.F.C. is funded by The São Paulo Research Foundation (FAPESP), grants
2022/09603-9 and 2023/14316-1.

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