Skip to main content

All Questions

Filter by
Sorted by
Tagged with
1 vote
0 answers
117 views

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
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
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
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
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
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
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
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
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
  • 4,145
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
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
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
  • 4,145
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
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
395 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
2 votes
0 answers
71 views

Domain of definition of a certain mapping

Suppose we have a compact smooth riemannian manifold $(M,g)$ and a $\mathcal{C}^2$ diffeomorphism $f$ of this manifold. I am studying the mapping $$\tilde{f}(x)= \exp_{f(x)}^{-1} \circ f \circ \exp_x \...
Giuseppe Tenaglia's user avatar
0 votes
1 answer
318 views

Estimating scalar curvature by norm of Riemannian curvature tensor under the Ricci flow

In B. Chow and D. Knopf's book "The Ricci Flow: An Introduction", the authors claim that for any dimension $n$ and any Riemannian manifold $M^n$, there is a constant $C_n$ depending only on $...
Matheus Andrade's user avatar
4 votes
0 answers
195 views

Classifying singularities of the Ricci flow

Context: A solution $(M^n, g(t))$ of the Ricci flow is said to encounter a Type III Singularity if $g(t)$ is defined for all $t \geq 0$ and: $$ \sup _{\mathcal{M}^{n} \times[0, \infty)} \|\...
Matheus Andrade's user avatar
2 votes
0 answers
354 views

Continuity of surface integrals on level sets

Let $\phi:\mathbb{R}^2\to\mathbb{R}$ such that $\phi^{-1}(0)\neq\emptyset$ and $\phi\in C^1(W)$ where $W$ is a compact neighborhood of $\phi^{-1}(0)$, with $\nabla\phi\neq 0$ in $W$. So there is some $...
Bogdan's user avatar
  • 1,759
13 votes
1 answer
654 views

Are there examples of Einstein manifolds with unbounded curvature?

Are there any known examples of Einstein manifolds $(M, g)$ such that $$\sup_{x \in M} \|\text{Rm}(x) \| = \infty$$ I'm looking for these examples because they might provide a counter-example to a ...
Matheus Andrade's user avatar
3 votes
0 answers
101 views

Minimal normal graph

Let $(M^3,g)$ be a complete and orientable Riemannian $3$-manifold and let $\Sigma^2 \subset M$ be a compact orientable totally geodesic surface embedded in $M$. For $f \in C^{2,\alpha}(\Sigma)$ with ...
Eduardo Longa's user avatar
4 votes
1 answer
248 views

A question on null geodesics in Lorentzian geometry

Suppose that $(M,g)$ is a smooth Lorentzian manifold, that $I\subset \mathbb R$ is a closed finite interval and that $\gamma: I \to M$ is a smooth null geodesic on $M$, that is to say, it satisfies $$ ...
Ali's user avatar
  • 4,145
2 votes
0 answers
35 views

Can one extend a Hermitian bundle from a compact manifold with boundary to its Riemannian double?

Let $M$ be a compact Riemannian manifold with boundary, and let $E \to M$ be a Hermitian vector bundle, endowed with a compatible connection. Let $\tilde M$ be a Riemannian double of $M$. Does $E$ ...
Alex M.'s user avatar
  • 5,407
0 votes
1 answer
108 views

Intersection Grassmanian planes

I am reading a paper that used Grassmanian planes properties. In particular, they studied the intersection of Grassmanian planes; they check the intersection Grassmanian of $n-k$-planes and ...
Adam's user avatar
  • 1,043
-4 votes
2 answers
303 views

Constructing a new manifold with a germ of manifold [closed]

Given a germ of manifolds and compatible Riemannian metrics, can we construct a new Hausdorff manifold using the exponential map? A germ of manifolds at a point $m$ is a series of manifolds $U_i$ ...
user avatar
6 votes
2 answers
428 views

An abstract characterization of line integrals

Let $M$ be a smooth manifold (endowed with a Riemann structure, if useful). If $\omega \in \Omega^1 (M)$ is a smooth $1$-form and $c : [0,1] \to M$ is a smooth curve, one defines the line integral of $...
Alex M.'s user avatar
  • 5,407
6 votes
1 answer
338 views

Is Gauss map of a free boundary convex disk a diffeomorphism?

I asked this question on MSE, but obtained no answer. Maybe this is the right place to post it. Let $D$ be a properly embedded free boundary disk in the closed unit ball $\mathbb{B}^3$ of $\mathbb{R}^...
Eduardo Longa's user avatar
2 votes
1 answer
95 views

conformal changes to Lorentzian curvature

Let $(M,g)$ be a Lorentzian manifold and let $R$ be the curvature tensor. We say $R\leq 0$ if $$ g(R(X,Y)Y,X) \leq 0\quad \forall \, X,Y \in TM.$$ My question is whether given a Lorentzian manifold $...
Ali's user avatar
  • 4,145
5 votes
1 answer
278 views

Levi-Civita connection from idempotents

Let $(M,g)$ be a closed Riemannian manifold. Let $V$ be a smooth complex vector bundle over $M$. We can write $V$ as the range of an idempotent $E$ in a matrix algebra $M_n(C^\infty(M))$ acting on a ...
geometricK's user avatar
  • 1,903
4 votes
1 answer
1k views

Geodesic convexity and the Geometric Hessian

This is an elementary question in differential geometry. We know that for a smooth real-valued function $f$ defined on an open geodesically convex set of a Riemannian manifold $ \mathcal{X} \subset \...
user1952770's user avatar
2 votes
0 answers
205 views

Can a non-compact manifold become compact by cutting it?

I'm trying to understand a step in a proof, where one starts with a non-compact manifold $V$ containing a trapped (2-sided, closed) surface $\Sigma$ that's non-separating. In order to complete the ...
aceituna's user avatar
  • 121
2 votes
0 answers
137 views

Heat-Flow on continuous differential forms and the Feller peroperty

Let (M,g) be a complete Riemannian manifold. It is well known that the Laplace operator is essentially self-adjoint on $C^\infty_c(M)$. This extends to the (de Rahm) Laplace operator on forms. Thus in ...
Nathanael Schilling's user avatar
0 votes
0 answers
466 views

Example metrics for exotic R4

I'm a physics student trying to understand what exotic manifolds, such as exotic R4, means. Is there known examples what the Riemannian metric of some exotic R4 (or some exotic sphere) would be? Does ...
Kirby's user avatar
  • 113
3 votes
0 answers
531 views

Geodesics (Local vs Global)

Let $M$ be a Riemannian manifold, and $p,q\in M$ two points. Now, if $M$ is a complete metric space the Hopf–Rinow theorem ensures that there is a geodesic $C\subset M$ joining $p$ and $q$ that ...
user avatar
6 votes
1 answer
310 views

Asymptotic bound on minimum epsilon cover of arbitrary manifolds

Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\varepsilon)$ denote the minimal cardinal of an $\varepsilon$-cover $P$ of $M$; ...
user141213's user avatar
1 vote
0 answers
123 views

Subdividing a Compact Bounded Curvature Manifold into Charts with Bounded Lipschitz Constant

Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$ cover $P$ of $M$; that is ...
user141213's user avatar