All Questions
Tagged with riemannian-geometry smooth-manifolds
171 questions
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 $$\cos\theta(w,v)=\frac{g_w(w,v)}{\sqrt{g_w(w,w)g_w(...
- 227
0
votes
0
answers
128
views
Is this possible to calculate norm of a vector using its inner product with another vector
Given a manifold $M$ with two Riemannian metrics $g_1$ and $g_2$ on $M$, a vector field $\xi$, and a smooth function $f:M\to \mathbb{R}$, is this possible to calculate $g_2(\xi,\xi)$ having the ...
- 227
2
votes
0
answers
124
views
How do conformal maps affect curvature?
Let $(\overline{M}^{n+1}, \langle \cdot, \cdot \rangle)$ be a riemannian manifold with riemannian connection $\overline{\nabla}$ and consider $M^n \subset \overline{M}$ an orientable hypersurface with ...
- 2,095
6
votes
0
answers
197
views
Regarding a proof in the surgery theorem by Gromov and Lawson
I have a question regarding a proof in the article The classification of simply connected manifolds of positive scalar curvature written by Gromov and Lawson. The precise reference is:
Gromov, ...
- 386
5
votes
1
answer
264
views
What are the minimal local models for Riemannian manifolds? A local question about isometric embeddings
There are many results about isometric embeddings of Riemannian manifolds but I haven't been able to find one that quite answers this question (which I believe must have some kind answer in the ...
- 7,789
1
vote
0
answers
225
views
Extending fibre metrics of submanifolds to Riemannian metrics
Let $M$ be a smooth manifold and $S\subseteq M$ a properly embedded smooth submanifold. Suppose that we have a fibre metric on $TM|_S$, i.e. a positive definite real inner-product on $T_pM$ for all $p\...
- 11
3
votes
1
answer
367
views
Is there such a connection on the punctured plane?
Is there a connection on $\mathbb{R}^2 \setminus \{0\}$ for which all operators of parallel transports are in the form $$\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
1
vote
1
answer
423
views
Riemannian Manifolds of Bounded Curvature
I am a complete newbie Riemannian Geometry with a particular application in mind so please excuse a lack of rigor in the question.
Suppose I have a manifold with sectional curvature everywhere ...
- 161
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
28
votes
3
answers
2k
views
Does isometric immersion map boundary to boundary?
Let $M$ be a compact, connected, oriented, smooth Riemannian manifold with non-empty boundary. Let $f:M \to M$ be a smooth orientation preserving isometric immersion.
Is it true that $f(\partial M) \...
- 6,741
7
votes
1
answer
373
views
Are metric isometries smooth at the boundary?
Let $M,N$ be smooth Riemannian manifolds with boundary (In particular, we assume the boundaries are smooth).
Suppose we have a map $\phi:M \to N$ which satisfies the following properties:
$$(1) \, \,...
- 6,741
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
12
votes
0
answers
381
views
Two ways a manifold can have little symmetry
Let $M$ be a closed connected smooth oriented manifold. The following two properties - that $M$ can either enjoy or not - intuitively both mean that $M$ has very little symmetry:
(a) Every self-map $...
- 11.9k
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
2
votes
0
answers
217
views
If $X$ is a compact smooth Riemannian manifold, why don't we integrate on a fundamental domain in the universal cover? [closed]
Let $X$ be a compact connected Riemannian manifold. The metric gives a local volume form. The universal cover is orientable, and has a precompact subspace locally isometric (with the covering metric) ...
- 73
25
votes
2
answers
1k
views
Is there a smooth manifold which admits only rigid metrics?
Does there exist a (finite dimensional) smooth manifold $M$, such that every Riemannian metric on $M$ has no isometries except the identity?
Of course, such a manifold must not admit a diffeomorphism ...
- 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
326
views
material derivative and relation to Riemannian metric
For each $n$ let $N_t$ be an embedded smooth hypersurface in $\mathbb{R}^n$ of dimension $n-1$. $\{N_t\}_t$ is a family of hypersurface that is evolving with some velocity $V$.
Smooth functions on $N$...
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
16
votes
1
answer
2k
views
Hodge de Rham operator and orientability
Let $(M,g)$ be a Riemannian manifold. One can consider the exterior algebra bundle $\Lambda(T^*M)$. The sections of this bundle are differential forms, to be noted by $\Omega^k(M)$. One can consider ...
- 9,330
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
4
votes
1
answer
434
views
Smooth manifolds for which every metric is geodesically convex
Are there non compact smooth manifolds which have the property that every Riemannian metric is geodesically convex?
Note that a manifold for which every Riemannian metric is complete must be compact.
...
- 6,741
1
vote
0
answers
485
views
Taylor expansion in Riemannian foliations
Take:
$M$ a Riemannian manifold, ${X_0}\in M$,
$N_{X_0}$ a submanifold of $M$ going through ${X_0}$,
and $Z \in N_{X_0}$ in a neighborhood of ${X_0}$.
At ${X_0} \in N_{X_0}$, we consider the ...
- 73
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
1
vote
1
answer
250
views
Global geometry measures for Riemannian manifolds
I'm working on a stochastic algorithm and considering it to apply in case of any curved space (manifolds). But in order to make the algorithm as efficient as possible I want to include in it some ...
- 267
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
1
vote
1
answer
213
views
Some quantities which definitions are (somehow) similar to the classical Divergence
Motivated by classical formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$ and $L_{X} \Omega=Div(X) \Omega$ and the essential role of the diff operator $d$ in definition of divergence, we define some ...
- 356
4
votes
1
answer
265
views
New setting for tensor definition
This is the standard definition for a tensor in a smooth manifold (Nakahara, for example):
But while reading Salamon's Riemannian Geometry and Holonomy Groups I have found this rather different ...
- 2,091
2
votes
1
answer
245
views
Divergence invariant lifting of a vector field via a submersion
What is an example of a smooth submersion $P:S^{3}\to S^{2}$ for which the following statment is Not true:
For every vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$...
- 356
16
votes
1
answer
600
views
If all balls around two points are isometric... -- manifold version
This question is a natural follow-up of this other question, asked earlier today by wspin.
Let's say that a metric space $(X,d)$ has two poles if:
there are two distinct points $x$, $y$ such that ...
- 10.9k
2
votes
0
answers
255
views
The Cauchy Problem in General Relativity: Existence of a Hausdorff Development
This is related to a problem that I posed about a year ago. I was given several references by a number of experts who were kind enough to entertain my rather arcane question. Those references were ...
- 307
0
votes
0
answers
140
views
Question about a particular estimate in Riemannian geometry
I have been studying the book Some Nonlinear Problems In Riemannian Geometry - Thierry Aubin. On page $46$ he begins the proof of the Sobolev imbedding theorem to manifolds. The proof is divided in ...
- 151
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