All Questions
Tagged with riemannian-geometry smooth-manifolds
171 questions
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
$$M=\mathbb{R}...
- 1,171
3
votes
1
answer
189
views
Randomly perturbed function has no accumulated critical point almost surely?
Given a smooth function $f$ and a smooth manifold $\mathcal{M}$ in $\mathbb{R}^d$, define the set
$$
S(v):=\{x:{\rm Proj}_{T_x{\mathcal{M}}}(v)=\nabla_{\mathcal{M}}f(x)\}.
$$
Is correct to say that $S(...
- 43
4
votes
1
answer
107
views
Identify an SDE on the sphere from its generator
I have a diffusion on the 2-sphere with expression:
$$
(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)
$$
...
- 73
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 ...
- 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 ...
- 808
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 $...
- 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}...
- 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$ ...
- 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 ...
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 ...
- 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 ...
- 1,219
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 ...
- 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 ...
- 659
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:
...
- 2,095
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{...
- 1,219
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(...
- 518
2
votes
1
answer
218
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$...
- 2,095
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)$ ...
- 2,095
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} \...
- 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 ...
- 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 ...
- 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}(\...
- 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 $C^{\infty}(\mathcal{M})$ ...
- 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 ...
- 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 ...
- 2,095
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 $...
- 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 $\left(W_t\right)_{t \geqslant 0}$ ...
- 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 $M$...
- 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 ...
- 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 ...
- 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 ...
- 2,095
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 ...
- 4,145
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 ...
- 431
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 ...
- 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 ...
- 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 $$\operatorname{Rm} = \begin{pmatrix} A & B \\...
- 659
3
votes
1
answer
486
views
There exists differentiable curves arbitrarily close to the continuous ones
Let $M$ be a Riemannian manifold; if $d$ is the distance on $M$, we can consider the distance $D$ between any two continuous curves given by $D(c, \gamma) = \max _{t \in [0,1]} d(c(t), \gamma(t))$.
...
- 5,407
6
votes
2
answers
435
views
The convex hull of a manifold whose cobordism class is trivial
Let $M$ be a compact orientable $n$ dimensional manifold. Assume that $M$ has trivial cobordism class.
Is there an embedding of $M$ in some Euclidean space $\mathbb{R}^m$ such that the convex ...
- 356
5
votes
0
answers
219
views
Is the volume functional analytic in the space of embeddings? What about locally?
Let $(M^{n+1},g)$ be an analytic Riemannian manifold and let $\Sigma^n$ be a closed analytic manifold. Denote by $\operatorname{Emb}(\Sigma, M)$ the space of all smooth (or maybe analytic) two-sided ...
- 2,095
3
votes
0
answers
117
views
Geometric intuition behind definition of $\delta$-necklike points of the Ricci flow
In "The Ricci Flow: An Introduction", the authors define a $\delta$-necklike point of the Ricci flow as a point $(x, t)$ where $$\|\text{Rm} - R (\theta \otimes \theta)\| \leq \delta \|\text{...
- 659
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 ...
- 4,081
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 ...
- 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 \...
- 985
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 ...
- 2,095
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 ...
- 356
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 ...
- 2,095
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 ...
- 203
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 $.
...
- 4,081
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 ...
- 4,081
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 ...
- 4,081