All Questions
103 questions
44
votes
5
answers
6k
views
Finding a 1-form adapted to a smooth flow
Let $M$ be a smooth compact manifold, and let $X$ be a smooth vector field of $M$ that is nowhere vanishing, thus one can think of the pair $(M,X)$ as a smooth flow with no fixed points. Let us say ...
- 114k
36
votes
3
answers
3k
views
When is a closed differential form harmonic relative to some metric?
Let $\omega$ be a closed non-exact differential $k$-form ($k \geq 1$) on a closed orientable manifold $M$.
Question: Is there always a Riemannian metric $g$ on $M$ such that $\omega$ is $g$-harmonic,...
- 585
26
votes
2
answers
4k
views
Questions on J. F. Nash's answer about his errors in the proof of embedding theorem
In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked
Is it true, as rumours have it, that
you started to work on the embedding ...
- 541
26
votes
2
answers
2k
views
Ellipses on spheres (and other surfaces)
Define an ellipse $E$ on a sphere as the locus of points whose sum of
shortest geodesic distances to two foci $p_1$ and $p_2$ is a constant $d$.
There are conditions on $\{ p_1, p_2, d \}$ for this ...
- 151k
20
votes
5
answers
2k
views
Smoothness of the closest point on a submanifold
Let $(M,g)$ be a smooth Riemannian manifold, and let $S \subseteq M$ be a compact submanifold.
Assume that for each $p \in M$, there exist a unique closest point on $S$, i.e a unique point $\tilde s(...
- 6,741
18
votes
3
answers
2k
views
Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order derivational operators
EDIT: According to some comments on this post I revise the title to remove the misunderestanding.
Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated ...
- 356
15
votes
3
answers
2k
views
Characterizing Hessians among symmetric bilinear tensors
I apologize in advance if this is somewhat elementary, but:
Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\in Sym^2(M)$ ...
- 5,891
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 ...
- 659
11
votes
5
answers
2k
views
Ricci Curvature in infinite dimensions?
Is there a good notion of "Ricci curvature" in infinite dimensions?
My intuitive understanding of Ricci curvature is that it is some kind of an "average" of the curvature tensor over "different ...
- 245
11
votes
1
answer
1k
views
Does the gluing procedure in Robert Wald’s book *General Relativity* yield a Hausdorff spacetime?
Before I state my problem, let me provide some definitions pertaining to the Cauchy Problem in General Relativity.
Definition 1: A triplet $ (\Sigma,h,k) $ is called an initial data set if $ (\Sigma,...
- 307
10
votes
2
answers
767
views
Intuition for the Drift Term of the Laplace-Beltrami Operator
In coordinates, the Laplace-Beltrami operator on a Riemannian manifold $(M,g)$ can be written as:
$$
\Delta_g = g^{ij}\partial_{ij} - g^{jk}\Gamma^\ell_{jk}\partial_\ell
$$
The second term:
$$
\mu^\...
- 213
9
votes
3
answers
752
views
Non-Kahler manifolds where the different Laplacians are compatible
On a Kahler manifold, the different Laplacians are compatible: $\Delta_d=2\Delta_{\bar{\partial}}=2\Delta_{\partial}$.
Are there non-Kahler Hermitian manifolds where the above identity holds?
user2529
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
696
views
Geodesics on manifolds with boundary
Let $(M,g)$ be a Riemannian manifold with non-empty boundary. Is there any notion of injectivity radius on $(M,g)$ in points away from the boundary? By this I mean points lying in $M- \partial M$. ...
- 131
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
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
7
votes
1
answer
259
views
Harmonic function with injective boundary conditions is an immersion?
Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are given an immersion $f:M \to \mathbb{R}^n$. (i.e $df$ is invertible at every point $p \in M$, note ...
- 294
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
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 $...
- 5,407
6
votes
2
answers
1k
views
Riemannian metrics preserved by diffeomorphisms
Let $f \neq Id$ be a diffeomorphism (of a smooth manifold $M$) which admits some Riemannain metric on $M$ making it an isometry. How many different metrics are preserved by $f$?
Note that $Met(f)=\{g|...
- 6,741
6
votes
1
answer
1k
views
Holonomy of a Kähler manifold
Hi,
I have the following question: Let $(M,J, \omega)$ be a Kähler manifold (not necessary compact). We know that the holonomy group is a subgroup of $U_{n}$. Let $\Omega$ be a constant ($\nabla \...
- 93
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
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}^...
- 2,095
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$; ...
- 71
5
votes
2
answers
377
views
Existence of an isotopy in Riemannian manifold
Let $(M,g)$ be a Riemannian manifold, and $p,q\in M$ be two fixed points. We assume $p,q$ are close enough. Say, we assume $p$ and $q$ are in the same normal coordinate chart. It is clear that there ...
- 2,789
5
votes
2
answers
7k
views
Inner products on differential forms
Given a Riemannian metric $g$ on a smooth manifold $M$, one defines an
$L^2$-inner product on the space $\bigwedge^\ast(M)$ of differential
forms by
$$
\langle \alpha, \beta \rangle_g = \int_M \...
- 585
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
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
5
votes
1
answer
264
views
What are the minimal local models for Riemannian manifolds? A local question about isometric embeddings
There are many results about isometric embeddings of Riemannian manifolds but I haven't been able to find one that quite answers this question (which I believe must have some kind answer in the ...
- 7,789
5
votes
1
answer
495
views
Volume comparison on Grassmannian
Let $G(r,n-r)$ be the Grassmannian, which can be identified with the space of all rank $r$ projection matrices in $\mathbb{R}^n$. Let $\mu$ be the uniform measure on $G(r,n-r)$. For any $\lambda_1,......
- 1,495
5
votes
1
answer
444
views
heat kernel on closed manifolds - error in Chavel's book?
first of all, I am not sure if this question fits here. I asked this question on math.stackexchange also but didn't get an answer so far.
In Isaac Chavel's book Eigenvalues in Riemannian Geometry, ...
- 53
5
votes
1
answer
817
views
Partitions of Unity
Fix a metric $g$ on a smooth, closed manifold $\mathcal{M}$. Take a finite subcover of the manifold from its atlas. Is it true that any smooth partition of unity subordinate to this cover has ...
- 1,123
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 ...
- 1,903
5
votes
1
answer
1k
views
Orthogonal complements in Hilbert bundles
It's a standard fact that for a finite-dimensional vector bundle with an inner product, the othogonal complement of any subbundle is itself a locally trivial vector bundle.
What is known about the ...
- 8,803
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
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
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 $.
...
- 4,081
4
votes
4
answers
3k
views
space of geodesics
hallo,
i have the following problem: Let $(M,g)$ be a compact Riemannian manifold with metric $g$ and $\nabla$ be the Levi-Civita Connection. Denote by $G(M) =${$\gamma: \mathbb{R} \rightarrow M | \...
- 213
4
votes
2
answers
575
views
Do transvers foliations induce complex structure?
Hallo,
I have the following question: Let $M$ smooth analytic manifold of dimension 4n. Assume furthermore that $M$ admits two foliations $A$, $B$, both with leaves of dimension 2n such that the ...
- 41
4
votes
1
answer
249
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
$$ ...
- 4,143
4
votes
2
answers
666
views
Unique Kahler-Einstein metric $g$ with $\mathrm{Ricc}(g)=-g$ when first Chern class $C_1(M)<0$: $\mathrm{Ricc}(h)=-g\,\Rightarrow\,h=cg$ for $c>0$?
On a compact Kahler manifold, let $g$ be the unique Kahler-Einstein metric with $\mathrm{Ricc}(g)=-g$, proved to exist by Yau and Aubin when the first Chern class $C_1(M)<0$.
Question: Does $g$ ...
- 376
4
votes
1
answer
797
views
The heat kernel as an exponential of an integral
In $\mathbb{R}^n$, if $\gamma$ is a line segment between $x_0 = \gamma (0)$ and $x = \gamma (t)$, one has the following formula:
$$\frac {\mathbb{e}^{- \frac{1}{4} \int_0^t <\dot{\gamma}, \dot{\...
- 5,407
4
votes
1
answer
137
views
Geodesic-like curves stemming from the heat kernel on a manifold
Consider a smooth $n$-dimensional Riemannian manifold $M$ with sufficiently nice geometric and topological properties such that there exist a unique heat kernel $(t,x,y) \mapsto p(t,x,y)$ on it ($t>...
- 5,407
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 \...
- 119
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
$$\Omega^{k}(\mathcal{M})\cong\mathcal{H}^{k}(\mathcal{M})\oplus\mathrm{ran}(\...
- 1,171
4
votes
0
answers
196
views
Let $p : \tilde{M} \to M$ be the universal cover. Can we ever deduce curvature properties of $M$ from the curvature of $\tilde{M}$?
Let $M$ be a $C^{\infty}$-smooth, connected, paracompact manifold with universal cover $\tilde{M}$. Assume $M$ is not simply connected, so that the covering map $p : \tilde{M} \to M$ is not the ...
- 1,379
4
votes
0
answers
195
views
Classifying singularities of the Ricci flow
Context:
A solution $(M^n, g(t))$ of the Ricci flow is said to encounter a Type III Singularity if $g(t)$ is defined for all $t \geq 0$ and:
$$
\sup _{\mathcal{M}^{n} \times[0, \infty)} \|\...
- 659
4
votes
0
answers
116
views
$\ell_p$ geodesic distance on smooth Riemannian manifold and Logarithmic Sobolev Inequalities
Bear with me, I'm not a professional geometer. Recently, I've been studying Logarithmic Sobolev Inequalities (LSI) for probability distributions on manifolds (e.g as done in works of Bobkvo et al. ...
- 6,853
4
votes
0
answers
191
views
A quantity associated with a Riemannian surface
Assume that $E$ is a Riemannian vector bundle, then its structure group is reduced to $O(n)$. Then the structure group of $E \oplus E$ is reduced to $D(O(n) \oplus O(n)) \subset Sp(2n)$ ...
- 356
3
votes
1
answer
778
views
Conformally-flat
Assume given a smooth manifold $(\mathbb{R}^n, g)$, where the metric is a scaled identity $g = e^{2f}I$.
Is there a way to know if this is always a non-positive (sectional) curvature manifold?
Note ...
- 75