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Question on globally hyperbolic manifolds and coordinates

Consider a globally hyperbolic Lorentzian manifold $(M,g)$. Then, a well-known result of Bernal-Sánchez (see Theorem 1.1 in arXiv:gr-qc/0401112) states that it can globally be written as $$M=\mathbb{R}...
B.Hueber's user avatar
  • 1,171
3 votes
1 answer
189 views

Randomly perturbed function has no accumulated critical point almost surely?

Given a smooth function $f$ and a smooth manifold $\mathcal{M}$ in $\mathbb{R}^d$, define the set $$ S(v):=\{x:{\rm Proj}_{T_x{\mathcal{M}}}(v)=\nabla_{\mathcal{M}}f(x)\}. $$ Is correct to say that $S(...
dkyopt's user avatar
  • 43
4 votes
1 answer
107 views

Identify an SDE on the sphere from its generator

I have a diffusion on the 2-sphere with expression: $$ (L\phi)(u):=\frac{1}{2{N(u)}}\Big(f(u)\Delta_{\mathbb S^2}\phi+ 2g\left( \nabla_{\mathbb S^2}\phi, \nabla_{\mathbb S^2}f\right)\Big) $$ ...
user3177306's user avatar
2 votes
0 answers
107 views

Finite dimensional manifolds as subspace of $\mathbb{R}^\mathbb{N}$

For embedded submanifold, specifically with ambient space being $\mathbb{R}^{n}$, there are many nice properties and results. Specifically there are many examples of matrix manifolds such as the ...
patchouli's user avatar
  • 275
8 votes
1 answer
230 views

The closure of the space of Riemannian metrics with a fixed isometry class

Let $M$ be a closed manifold, and let $\mathscr{M}$ be the space of all Riemannian metrics over $M$. It is known that this is a Fréchet manifold. Consider also $\mathscr{D}$ the diffeomorphisms group ...
MyShepherd's user avatar
1 vote
1 answer
182 views

Lower bound on injectivity radius at one point implies lower bound on injectivity radius for a closed manifold

I’m interested in a closed Riemannian manifold $(M^n,g)$ with $sec<0$ and $diam(M)\leq D$. My question is: If at some point $p\in M^n$, the injectivity radius $injrad(p)\geq1$, then can we get $...
Xin Qian's user avatar
  • 155
0 votes
0 answers
35 views

Charecterizing (Riemannian) submanifolds of the Bures-Wasserstein manifold

I'm still learning Riemannian geometry, so please correct any mistakes. I am interested in the Bures-Wasserstein manifold of centered Gaussians of dimension $d$. In this case, the manifold $\mathcal{M}...
Afham's user avatar
  • 41
1 vote
0 answers
88 views

Metric of negative curvature on connected sum

Let $(M_1,g_1)$ and $(M_2,g_2)$ be two Riemannian manifolds of dimension $n\geq 2$. If we consider the connected sum $M=M_1\mathbin{\#}M_2$ of the two manifolds; can one get a smooth metric $g$ on $M$ ...
User5's user avatar
  • 11
2 votes
0 answers
81 views

Assumptions for uniform measure of SDE on manifolds

Suppose we're working on a compact, Riemannian manifold $M$. Suppose $dX_t = -b(X_t, t)\,dt + \sigma^2 \,dB_t$ is started at the uniform measure on $M$. What kind of assumptions on $b$ make it so that ...
optimal_transport_fan's user avatar
5 votes
1 answer
343 views

Clarifying a result of Klingenberg

I am looking for some clarification on a result of Klingenberg. For context, I have a complete Riemannian manifold for which I would like to compute a lower bound on the injectivity radius at each ...
E G's user avatar
  • 163
0 votes
1 answer
202 views

How to estimate the distance between geodesics and points for Riemannian manifold with positive sectional curvature

Assume that $ M $ is a complete Riemannian manifold and there exists $ k>0 $ such that $ K(q)\geq k $ for any $ q\in M $, where $ K $ is the sectional curvature of $ M $. Let $ \gamma $ be a closed ...
Luis Yanka Annalisc's user avatar
6 votes
1 answer
248 views

Existence of adjoint operators on manifolds

Let $(M,g)$ be an oriented Riemannian manifold and $V$ a finite-rank vector bundle equipped with a non-degenerate bundle metric $\langle\cdot,\cdot\rangle_{V}$. This bundle metric, in turn, gives rise ...
G. Blaickner's user avatar
  • 1,429
2 votes
1 answer
104 views

Why is this subset associated to a $2$-tensor dense?

Let $S$ be a symmetric $(0, 2)$ tensor on a Riemannian manifold $M$. Define $E_S : M \to \mathbb{Z}$ by $E_S(x) = \left(\text{the number of distinct eigenvalues of } S_x\right)$. I've seen the ...
Matheus Andrade's user avatar
1 vote
1 answer
90 views

Infimum of the normalized Laplacian eigenvalues

Let $(M^n,g)$ be a compact Riemannian manifold. The spectrum of the Laplacian operator $\Delta_g = -\operatorname{div} \nabla$ consists of an increasing and diverging sequence of positive eigenvalues: ...
Eduardo Longa's user avatar
5 votes
1 answer
119 views

A formula in harmonic heat flow

Assume that $ (M,g) $ and $ (N,h) $ are two smooth closed manifold and $ N $ is embedded isometrically into $ \mathbb{R}^K $ for some $ K\in\mathbb{Z}_+ $. Assume that $ u\in C^{\infty}(M\times\mathbb{...
Luis Yanka Annalisc's user avatar
1 vote
1 answer
207 views

Is squared geodesic distance a Morse-Bott function on simply-connected Hadamard manifolds?

Let $M$ be a simply connected Hadamard manifold. That is, $M$ is a complete Riemannian manifold with sectional curvature bounded above by 0. Let $f(x,y)=\frac{1}{2}d^2(x,y)$, where $x,y \in M$, and $d(...
Spencer Kraisler's user avatar
2 votes
1 answer
217 views

Are these the only first eigenfunctions on a hemisphere?

Let $\mathbb{S}^2_+$ denote the closed upper hemisphere of the unit round sphere in $\mathbb{R}^3$. It is well known that the first positive eigenvalue of the Laplacian on the closed unit sphere is $2$...
Eduardo Longa's user avatar
2 votes
0 answers
44 views

$1$-parameter family of metrics preserving the normal direction

Let $(M^n,g)$ be a compact Riemannian manifold with boundary, $n \geq 2$, and let $N$ be the unit outward normal to $\partial M$. Denote by $S^2(M)$ the symmetric covariant $2$-tensors, by $S^2_0(M)$ ...
Eduardo Longa's user avatar
3 votes
0 answers
214 views

Implicit function theorem in Riemannian manifold and Wasserstein space

My question is about to what extent can we extend the implicit function theorem to Riemannian manifolds. In the Euclidean space, consider a bivariate function $F \colon \Theta \times \mathcal{X} \...
Steve's user avatar
  • 1,127
4 votes
1 answer
167 views

Is the intersection of such a triple of minimal surfaces in the 3-ball a single point?

Let $S_1,S_2,S_3$ be three simple closed curves on the 2-sphere $\mathbb{S}^2$. (With no smoothness or rectifiability assumption) For each $i$, let $M_i$ denote the minimal surface (i.e. disc) bounded ...
Agelos's user avatar
  • 1,926
6 votes
1 answer
423 views

Difference between parallel transport and ambient projection

Consider a $d$-dimensional complete embedded Riemannian submanifold $(M,g)$ of a Euclidean space $\mathbb{R}^D$ (The major examples we consider are sphere and Stiefel manifold). Assume the sectional ...
Jason Li's user avatar
  • 125
4 votes
0 answers
334 views

Hodge decomposition on non-compact manifolds

Let $(\mathcal{M},g)$ be a compact Riemannian manifold without boundary. Then we have the well-known Hodge decomposition $$\Omega^{k}(\mathcal{M})\cong\mathcal{H}^{k}(\mathcal{M})\oplus\mathrm{ran}(\...
B.Hueber's user avatar
  • 1,171
5 votes
3 answers
620 views

Poisson equation on manifolds

Let $(\mathcal{M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well-known that the Poisson equation $$\Delta u=f$$ does have a solution on $C^{\infty}(\mathcal{M})$ ...
B.Hueber's user avatar
  • 1,171
1 vote
0 answers
86 views

Poisson equations for tensors on compact Riemannian manifold

Let $({M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well known that the Poisson equation $$\Delta f=S$$ where $\Delta:C^{\infty}({M})\to C^{\infty}({M})$ denotes ...
B.Hueber's user avatar
  • 1,171
4 votes
1 answer
169 views

What integral formula is being used here?

I am trying to read the paper "Simple closed geodesics on convex surfaces" by E.Calabi and J. Cao and a certain passage is unclear for me. Before, let me contextualize and set up some ...
Eduardo Longa's user avatar
1 vote
1 answer
206 views

Decomposition of tensor field on hypersurface

Let $(\mathcal{M},g)$ be a Lorentzian manifold, which is globally of the form $\mathcal{M}\cong I\times\Sigma$, where $I\subset\mathbb{R}$ ("time") and $\Sigma$ ("space") is some $...
B.Hueber's user avatar
  • 1,171
2 votes
1 answer
483 views

What is an analogous version of the Ornstein–Uhlenbeck process on Riemannian manifolds?

Recall that the Ornstein–Uhlenbeck (OU) process in $\mathbb{R}^d$ is defined by the following SDE, $$ d Z_t=\frac{-1}{2} Z_t d t+d W_t, \quad t \geqslant 0 $$ where $\left(W_t\right)_{t \geqslant 0}$ ...
Student's user avatar
  • 537
4 votes
2 answers
364 views

Equivalence between two Sobolev norms on manifolds

On a compact Riemannian manifold $(M,g)$ without boundary, there are two ways to define a Sobolev norm on $M$. Assume that $f\in C^\infty(M)$ in the following. Use pseudo-differential operators on $M$...
kuuga's user avatar
  • 71
4 votes
0 answers
196 views

Let $p : \tilde{M} \to M$ be the universal cover. Can we ever deduce curvature properties of $M$ from the curvature of $\tilde{M}$?

Let $M$ be a $C^{\infty}$-smooth, connected, paracompact manifold with universal cover $\tilde{M}$. Assume $M$ is not simply connected, so that the covering map $p : \tilde{M} \to M$ is not the ...
AmorFati's user avatar
  • 1,379
3 votes
1 answer
332 views

Screened Poisson equation

The screened Poisson equation, i.e. $$ [\nabla^2−\lambda^2]\phi(r) = −\psi(r),$$ occurs frequently in physics, including Yukawa theory of mesons, in electric field screening in plasmas and nonlocal ...
MathDG's user avatar
  • 272
6 votes
1 answer
129 views

On properties of Besse spheres

Let $(\mathbb{S}^2,g)$ be a Besse sphere, that is, a Riemannian sphere all of whose geodesics are closed. By a result of Gromoll and Grove, all the geodesics are simple (no self-intersections) and ...
Eduardo Longa's user avatar
1 vote
1 answer
164 views

A question on convexity and conjugate points

Let $(M,g)$ be a compact smooth simply connected Riemannian manifold with a smooth boundary. Assume also that $(M,g)$ does not have any conjugate pairs of points. Let $\Gamma \subset \partial M$ be a ...
Ali's user avatar
  • 4,135
3 votes
1 answer
257 views

Space filling curves

The classic Hahn-Mazurkiewicz theorem has the following consequence: Let $X$ be a compact, connected topological manifold. Then there is a continuous surjective map $f: [0,1] \rightarrow X$. It is ...
Rahul Sarkar's user avatar
4 votes
0 answers
227 views

To what extent is the Nash embedding not unique?

Consider a smooth Nash embedding, $f$, of a Riemannian manifold $Σ$ into Euclidean space $\mathbb R^n$. To what extent is this embedding not unique? It is clear that the set of all such embeddings ...
dennis's user avatar
  • 521
2 votes
2 answers
387 views

Under what conditions can an orientable Riemannian 3-manifold be defined implicitly?

Under what conditions can an orientable Riemannian 3-manifold $\Sigma$ be defined implicitly? What I mean by implicitly is that there exists a smooth function $f:\mathbb{R}^n\to \mathbb{R}^m$, such ...
dennis's user avatar
  • 521
3 votes
1 answer
168 views

Curvature estimate in Hamilton's Ricci flow paper for traceless $\operatorname{Rm}$ on $4$-dimensional manifold

In dimension $4$, it is known that the curvature operator $\operatorname{Rm} : \Lambda^2(M) \to \Lambda^2(M)$ admits a block decomposition of the form $$\operatorname{Rm} = \begin{pmatrix} A & B \\...
Matheus Andrade's user avatar
3 votes
1 answer
486 views

There exists differentiable curves arbitrarily close to the continuous ones

Let $M$ be a Riemannian manifold; if $d$ is the distance on $M$, we can consider the distance $D$ between any two continuous curves given by $D(c, \gamma) = \max _{t \in [0,1]} d(c(t), \gamma(t))$. ...
Alex M.'s user avatar
  • 5,407
6 votes
2 answers
435 views

The convex hull of a manifold whose cobordism class is trivial

Let $M$ be a compact orientable $n$ dimensional manifold. Assume that $M$ has trivial cobordism class. Is there an embedding of $M$ in some Euclidean space $\mathbb{R}^m$ such that the convex ...
Ali Taghavi's user avatar
5 votes
0 answers
219 views

Is the volume functional analytic in the space of embeddings? What about locally?

Let $(M^{n+1},g)$ be an analytic Riemannian manifold and let $\Sigma^n$ be a closed analytic manifold. Denote by $\operatorname{Emb}(\Sigma, M)$ the space of all smooth (or maybe analytic) two-sided ...
Eduardo Longa's user avatar
3 votes
0 answers
117 views

Geometric intuition behind definition of $\delta$-necklike points of the Ricci flow

In "The Ricci Flow: An Introduction", the authors define a $\delta$-necklike point of the Ricci flow as a point $(x, t)$ where $$\|\text{Rm} - R (\theta \otimes \theta)\| \leq \delta \|\text{...
Matheus Andrade's user avatar
14 votes
1 answer
860 views

Mapping torus of Klein bottle

This got 5 upvotes but no answers on MSE (Mapping torus of Klein bottle), so I'm cross-posting to MO: The mapping torus of a Klein bottle $ K $ is a compact flat 3 manifold. The mapping class group of ...
Ian Gershon Teixeira's user avatar
3 votes
0 answers
109 views

Application of Santalo’s formula

Suppose that $(M,g)$ is a compact smooth Riemannian manifold with a smooth boundary and suppose that $f$ is a smooth function on $M$ with the property that $$ \int_I f(\gamma(t))\,dt=0,$$ for any ...
Ali's user avatar
  • 4,135
1 vote
1 answer
137 views

Smoothness of the asymptotic parametrization of a ruled surface

Let $S$ be a smooth developable surface in $\mathbb{R}^{3}$. It is well known that, if $S$ is free of planar points, then it admits a local parametrization of the form $$\begin{align} \sigma \colon I \...
Matteo Raffaelli's user avatar
5 votes
0 answers
101 views

How is this product of tensors defined?

I am reading the paper “ The first eigenvalue of a small geodesic ball in a riemannian manifold”, by Karp and Pinsky, from where I took the following: Here, $\Delta_{-2}$ denotes the usual Laplacian ...
Eduardo Longa's user avatar
2 votes
1 answer
256 views

Equidistant points on a compact Riemannian manifold

Let $(M,g)$ be a compact Riemannian manifold. To this Riemannian manifold, we associate a natural number $K(M,g)$ as follows: $K(M,g)$ is the maximum of all $n\in \mathbb{N}$ such that we have at ...
Ali Taghavi's user avatar
3 votes
1 answer
234 views

Global choice of eigenvectors on an open surface

Let $(M^2,g)$ be a noncompact orientable Riemannian surface without boundary. Let $A \in \Gamma(\operatorname{Sym}(TM))$ be a section of the bundle of symmetric endomorphisms of $TM$, that is, for ...
Eduardo Longa's user avatar
1 vote
0 answers
218 views

Is this generalization of differentiable manifolds to mixed dimensions a known object?

Suppose you are studying the evolution of some electromagnetic quantity in a conductor consisting of objects of several dimensions, i.e. wires, plates and balls. This would amount to studying the ...
Robert Wegner's user avatar
5 votes
1 answer
394 views

Embedding round manifolds into low dimensional spheres

Robert Bryant's answer to Isometric embedding of SO(3) into an euclidean space mentions that there is an isometric embedding of the round tetrahedral space $ SO_3/A_4 $ into the round sphere $ S^6 $. ...
Ian Gershon Teixeira's user avatar
9 votes
0 answers
336 views

Nash embedding for 3 manifolds

The Nash embedding theorem tells us that every smooth Riemannian m-manifold can be embedded in $R^n$ for, say, $n = m^2 + 5m + 3$ (edit: 14 is a better bound for compact 3 manifolds thanks @mme). What ...
Ian Gershon Teixeira's user avatar
8 votes
1 answer
599 views

Exact condition for smooth homogeneous to imply Riemannian homogeneous for compact manifolds

Let $ (M,g) $ be a homogeneous Riemannian manifold. That is, the isometry group $ Iso(M,g) $ acts transitively on $ M $. Let $ \pi_1(M) $ be the fundamental group of $ M $. Then $ \pi_1(M) $ has ...
Ian Gershon Teixeira's user avatar