All Questions
103 questions
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
0
votes
0
answers
90
views
The idealizer of the space of vector fields with vanishing divergence
The space of smooth vector fields on a manifold is counted as a Lie algebra with the usual Lie bracket structure.
Is there a Riemannian manifold of dimension at least $2$ which satisfies either of ...
- 356
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
2
votes
0
answers
216
views
A geometric rank of Riemannian manifolds
There are various ranks which have been assigned to a smooth manifold. The following ranks are two examples of such ranks:
The maximum number of global independent vector fields which can be defined ...
- 356
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
1
vote
0
answers
54
views
Extension of a function from a subset of a manifold to the unit sphere
There is a line in this following paper (page no- 221, the paragraph before the Lemma 2.2)
Otsu, Yukio; Shiohama, Katsuhiro; Yamaguchi, Takao, A new version of differentiable sphere theorem, Invent. ...
- 930
2
votes
1
answer
720
views
Riemannian manifolds: every compact subset is contained in a connected relatively compact open subset [closed]
While working on some problem (not relevant here), it turned out to be convenient to be able to enclose arbitrary compact subsets in "nicer" compact subsets, hence the question:
if $(M,g)$ is a ...
- 5,407
-2
votes
1
answer
112
views
Bochner theorem for complete manifolds
Let $(M,g)$ be a complete oriented Riemannian manifold. If $Ric ≥ 0$ on $M$, then any harmonic 1-form $\alpha$ is parallel i.e $x\to |\alpha|^2_{g,x}$ constant?
- 17
2
votes
1
answer
126
views
Stokes theorem outside of divisor
Let $X$ be a compact Kähler manifold and $D$ be a snc divisor on it. Then on $X\setminus D$ Stokes theorem holds true
$\int_{X\setminus D}\Delta\alpha=0\; \; \; ?$
In this case $X\setminus D$ is non-...
- 17
0
votes
0
answers
236
views
Angle between two vectors in a Minkowski (Finsler) space
Given a Minkowski (or Finsler) space $(V,F)$, I am wondering how to define the angle between two vectors $w$ and $v$. I first thought it must be as $$\cos\theta(w,v)=\frac{g_w(w,v)}{\sqrt{g_w(w,w)g_w(...
- 227
5
votes
1
answer
264
views
What are the minimal local models for Riemannian manifolds? A local question about isometric embeddings
There are many results about isometric embeddings of Riemannian manifolds but I haven't been able to find one that quite answers this question (which I believe must have some kind answer in the ...
- 7,789
1
vote
0
answers
225
views
Extending fibre metrics of submanifolds to Riemannian metrics
Let $M$ be a smooth manifold and $S\subseteq M$ a properly embedded smooth submanifold. Suppose that we have a fibre metric on $TM|_S$, i.e. a positive definite real inner-product on $T_pM$ for all $p\...
- 11
3
votes
1
answer
367
views
Is there such a connection on the punctured plane?
Is there a connection on $\mathbb{R}^2 \setminus \{0\}$ for which all operators of parallel transports are in the form $$\begin{pmatrix}a&-b\\b&a \end{pmatrix}$$
but the parallel ...
- 356
44
votes
5
answers
6k
views
Finding a 1-form adapted to a smooth flow
Let $M$ be a smooth compact manifold, and let $X$ be a smooth vector field of $M$ that is nowhere vanishing, thus one can think of the pair $(M,X)$ as a smooth flow with no fixed points. Let us say ...
- 114k
10
votes
2
answers
767
views
Intuition for the Drift Term of the Laplace-Beltrami Operator
In coordinates, the Laplace-Beltrami operator on a Riemannian manifold $(M,g)$ can be written as:
$$
\Delta_g = g^{ij}\partial_{ij} - g^{jk}\Gamma^\ell_{jk}\partial_\ell
$$
The second term:
$$
\mu^\...
- 213
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
0
votes
2
answers
535
views
Lie algebra of Gradient vector fields(2)
Motivated by this question, is there a $n$ dimensional Riemannian manfold $M$, $n>1$ such that the space of all gradient vector fields is a Lie algebra under the usual Lie bracket ...
- 356
2
votes
1
answer
429
views
The Lie algebra of gradient vector fields
Assume that $M$ is a differentiable manifold. Is there a Riemannian metric on $M$ such that the space of all gradient vector fields on $M$ would be closed under the Lie bracket?
- 356
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
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
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
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
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
1
vote
0
answers
84
views
A problem of defining addition in a Quotient space
Let $\mathcal{C}$ be the space of all parametric curves $x:[0,1]\rightarrow \mathbb{R}^2$. Let the set of all re-parameterizations of curves is $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma (...
- 213
1
vote
1
answer
321
views
Soliton equation and non-killing potential vector field
I am searching for a non-Killing vector field $\zeta \in\frak X\rm (M)$ where $(M,g)$ is a Riemannian manifold such that
$$\frac12 \frak L_\zeta \rm g+Ric=\lambda g$$
$$ \frak L_\zeta \rm Ric=\lambda \...
- 422
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
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
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
2
votes
1
answer
420
views
Nowhere vanishing, normalized vector field with bounded derivatives
It is well-known that any non-compact manifold admits a nowhere vanishing vector field. If we have a Riemannian metric we may pick such a vector field and normalize it so that at every point it has ...
- 2,998
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
1
vote
1
answer
348
views
Linearisation of Einstein operator
Let $(M,g)$ be a $(m+1)$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$.
The Ricci curvature can be viewed as a differential operator $\text{Rc}:\Gamma(S^2_+M)\rightarrow\...
- 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
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
1
vote
1
answer
237
views
A special type of transitivity
Let $M$ be a smooth orientable manifold with volume form $\Omega$. Fix two pints $x,y \in M$. Put $A$=all volume preserving diffeomorphism of M which maps $x$ to $y$.
Define $B$=All linear volume ...
- 356
1
vote
0
answers
127
views
Riemann normal coordiantes and change of metric
Le $(M,g)$ be Riemannian manifold. Fix point $p\in M$. We can define the map
$$\exp: U \subset T_p M \rightarrow M$$
$$\exp(X) = \gamma_{p,X}(1)$$
where $t\mapsto \gamma_{p,X}(t)$ is geodesic such ...
- 552
1
vote
1
answer
510
views
The space of generalized complex structures in sense of N.Hitchin is contractible?
Generalized complex structures were introduced by Nigel Hitchin in 2002. A generalized almost complex structure is an almost complex structure of the generalized tangent bundle which preserves the ...
user21574
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
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
3
votes
2
answers
669
views
Real analytic submanifolds of $\mathbb{R}^{n}$
Hallo,
Let $(M,g)$ be a Riemannian $k$-dim real analytic submanifold of $\mathbb{R}^{n}$. Is it true that $M$ in $\mathbb{R}^{n}$ looks locally (in a small neigbourhood around some point in $M$) as ...
- 339
3
votes
3
answers
1k
views
Rotation in Hyperkähler manifolds
Any Hyperkähler manifold has 3 complex structures $I_{1}, I_{2}, I_{3}$. Assume that there is an additional complex structure $J$. Can this be written as $J = aI_{1} + bI_{2} + cI_{3}$, where $(a,b,c) ...
- 339
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
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
2
votes
1
answer
425
views
holomorphic extension of forms
hallo,
I have the following question: Let $M$ be a $n-$dimensional complex manifold and $X \subset M$ be a compact $n-$dimensional totally real analytic Riemannian submanifold. Let furthermore $\...
- 29
0
votes
1
answer
194
views
relation with jacobifields in a small neighbourhood
hi,
I have the following question: Let $(M,g)$ be a complete Riemannian manifold with all sectional curvatures non-positive. Let $p \in M$ and consider the function $d(x)=dist_{g}(x,p)$ in a ...
- 89
1
vote
1
answer
397
views
Einstein metrics on the tangent bundle
Let $M$ be a compact, real analytic, riemannian manifold with real analytic metric $g$. Does the tangent bundle always admit an Einstein metric ?
- 213
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
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
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
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
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