All Questions
Tagged with dg.differential-geometry riemannian-geometry
1,985 questions
5
votes
1
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315
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Positively curved Riemannian manifolds
Let $M$ be a compact Riemannian manifold with positive sectional curvature whose universal covering space is diffeomorphic to $S^n$. Is $M$ diffeomorphic to a spherical space form?
I know, by a ...
- 377
0
votes
2
answers
374
views
On the definition of convergence of a sequence of sections of a bundle
Convergence of a sequence of sections of a bundle is defined as follows:
Definition: Let $E$ be a vector bundle over a manifold $M$, and let metrics $g$ and connections $∇$ be given on $E$ and on $TM$...
- 1,303
5
votes
1
answer
299
views
Can an open manifold with positive Ricci curvature be non simply connected at infinity?
The question is in the title, I haven't been able to locate a discussion of these kind of properties.
- 4,123
4
votes
1
answer
699
views
Spectrum of the Laplace-Beltrami operator on $L^p$: where is it?
On a noncompact Riemannian manifold $M$, the $L^2$-spectrum of the Laplace-Beltrami operator $\Delta$ sits inside $\mathbb{R}$ (by self-adjointness), either to the left or to the right of $0$ ...
user14166
2
votes
1
answer
490
views
Curve on a surface defined by its geodesic curvature
Suppose that $S$ is a smooth complete surface, and $c\colon [0,L]\to S$ is a smooth curve in $S$, parametrized by arc-length. Then $c$ is uniquely determined by its initial tangent vector and its ...
22
votes
1
answer
966
views
Avoiding integers in the spectrum of the Laplacian of a Riemann surface
Let $\Sigma^2$ be an orientable compact surface of genus $gen(\Sigma)\geq2$, and denote by $\mathcal M(\Sigma)$ the moduli space of hyperbolic metrics on $\Sigma$, i.e., Riemannian metrics of constant ...
- 5,891
2
votes
1
answer
1k
views
How to understand two examples of spin bundle
I am confused by two examples of spinor bundles over 4-manifolds, which I saw in various places:
(1) The spinor bundle $S = S_+ \oplus S_-$ associated to a spin or spinc structure of Riemannian four-...
- 857
3
votes
1
answer
2k
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What are Euler density and Weyl invariants?
I would like to know as to what is the definition and significance of what are called "Euler density" and "Weyl invariants" (of weight $-d$ on a $d-$manifold)
Do many (which?) of them vanish when ...
- 1,893
4
votes
1
answer
944
views
Interpetation of torsion and curvature in terms of families of nearby geodesics
Let $M$ be a Riemannian manifold with affine connection such that the metric is covariantly constant (so that the connection equals the Levi-Civita connection up to torsion).
I know the ...
- 2,399
0
votes
1
answer
738
views
Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?
Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me:
Is there any open Ricci-flat ALE 4-manifold other than ...
- 1,236
1
vote
0
answers
463
views
Reference request for parallel transport
I am learning about parallel transport on a Riemannian manifold equipped with an affine connexion. It seems (if I understand it well) that, in general, we might not be able to compute the parallel ...
- 119
11
votes
2
answers
687
views
Does positively curved sphere admit an isometric embedding as hypersurface in Euclidean space?
Let $(S^n, g)$ be an $n$-dimensional positively curved sphere. Assume the smoothness of the metric, does it admits an isometric embedding into $\mathbb R^{n+1}$?
for $n=2$ it is proved by A.D ...
- 2,623
9
votes
2
answers
2k
views
Does every manifold have a flat connection?
Suppose I have a manifold and a vector bundle over it, but not a connection or a metric. Can I always find a connection on it that has a Riemann curvature tensor that is identically zero? If so, can I ...
- 1,491
7
votes
1
answer
975
views
Reverse Ricci Flow and Longtime Existence
The usual Ricci flow and normalized Ricci flow for surfaces are
$$ \partial_t g = -2Kg $$
and
$$ \partial_t g = -2Kg + 2sg,$$
where $K$ is the Gaussian curvature and $s$ is its average.
The latter ...
- 355
18
votes
3
answers
4k
views
Formal adjoint of the covariant derivative
Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric.
It is essentially ...
- 5,824
12
votes
1
answer
2k
views
Isometry group of pseudo Riemannian manifold always a Lie group? (Myers-Steenrod)
Myers-Steenrod states that the isometry group of a Riemannian manifold is a Lie group. Is that also true for pseudo Riemannian manifolds? I didn't find anything related to that.
Cheers
- 123
4
votes
1
answer
1k
views
Regularity of metric of the double of a Riemannian manifold
Let $M$ be a Riemannian manifold with totally geodesic boundary $\partial M$. We let $\check{M}$ be its double, i.e. the disjoint union of $M$ with itself under identification of corresponding ...
- 105
1
vote
2
answers
287
views
Number of geodesics of certain length
Let $M$ be a Riemannian manifold, and let $x, y \in M$ be non-conjugate points.
Let $r, R>0$ be two numbers. I am looking for a bound on the number of geodesics between $x$ and $y$ of Length ...
- 9,853
0
votes
1
answer
283
views
Buseman function on manifolds with $Ric \ge - \left( {n - 1} \right)$
It's well known that if M is a Riemannian manifold with $Ric \ge 0$ and contains a line $\gamma $. Set ${\gamma _ + } = \gamma \left| {_{[0, + \infty )}} \right.$, ${\gamma _ - } = \gamma \left| {_{[ -...
- 199
6
votes
1
answer
868
views
Shortest geodesic loop vs. shortest periodic geodesic
Are there simple conditions on a Riemannian metric on the two-sphere that imply that a geodesic loop of minimal length is actually a periodic geodesic?
For example, is this true for small ...
- 13.5k
7
votes
2
answers
313
views
Isometric embedding as a graph
Question
Let $M$ be a (finite dimensional) smooth manifold and $g,\bar{g}$ be Riemannian metrics on $M$.
Under what conditions can we guarantee that there exists another finite dimensional Riemannian ...
- 39k
13
votes
3
answers
5k
views
Can the hyperbolic plane be immersed in three dimensional Euclidean space, if we are only looking for a weak solution?
Consider the following question:
"Can the hyperbolic plane $(\mathbb{R}^2, g_H)$ be isometrically
immersed in three dimensional Eulidean space$(\mathbb{R}^3, g_{flat})$?"
I believe the answer to ...
- 3,245
0
votes
1
answer
257
views
volume of a submanifold implies bounds on curvature
I would like to ask the following question: Suppose an m-dimensional manifold in an n-dimensional euclidean space, choose some point on this manifold and take an n-dimensional ball of radius R centred ...
5
votes
1
answer
613
views
Proof of the general expression for anomaly in a CFT and its partition function
I think the statement is that for any dimensional CFT the following is true,
$$\langle T^{\mu}_\mu \rangle = \sum B_n I_n - 2(-1)^{d/2}AE_d,$$
where $E_d$ is the `"Euler density" and $I_n$ are the ...
- 1,893
5
votes
1
answer
457
views
Can one use the continuity method to show that the two dimensional hyperbolic space can be immersed in five dimensional Euclidean space?
First of all, I must clarify at the outset that I am simply asking if there is an alternative way to solve an already known problem. It is known that the answer to my question is yes. The problem is ...
- 3,245
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
2
votes
1
answer
165
views
The measure on the harmonic spectrum from Selberg trace formula
One can see the following two equations,
Theorem 6.1 (Selberg Trace formula) on page 26 of these notes.
Equation 3.19 and 3.20 on page 11 of this paper.
I vaguely feel that these two are the same ...
- 1,893
1
vote
2
answers
674
views
Non simply connected HyperKähler 4-manifolds without ALE metrics
In a 1989 paper Peter Kronheimer showed that each simply connected HyperKähler 4-manifold possesses an ALE metric. What do we know about the non-simply connected cases?
- 1,236
0
votes
1
answer
451
views
Non-Symmetric Equivariant Riemannian Metrics on Homogeneous Spaces
For a homogeneous space $M = G/H$, the number of $H$-equivariant Riemannian metrics on $M$ is usually much smaller than the space of Riemannian metrics. I am wondering what happens when the symmetric ...
- 173
3
votes
0
answers
266
views
Dimensional curvature identities
In a series of papers (1, 2, 3) P. Gilkey et al. discuss certain identities satisfied by the curvature tensor of a (pseudo)-Riemannian metric.
Contrary to the Bianchi or Ricci identities, these ones ...
- 1,064
1
vote
1
answer
305
views
On the canonical neighborhoods
Suppose $M$ is a 3-dimensional manifold, John W. Morgan and Frederick Tsz-Ho Fong in their "Ricci Flow
and Geometrization
of 3-Manifolds" book as a definition of canonical neighborhoods have ...
- 1,303
11
votes
1
answer
540
views
Minimum requirements for a Kähler manifold to be hyperkähler
In 'panoramic view of Riemmannian geometry' when introducing hyperkähler manifolds, Berger states, informally, that a hyperkähler manifold is a Riemmannian manifold which is Kähler for more than one ...
- 4,123
8
votes
2
answers
589
views
Easy proof of topological property of Zoll manifolds
It is known that the cohomology ring of a Zoll manifold---a riemannian manifold all of whose geodesics are periodic with the same minimal period---must be the same as the cohomology ring of a compact ...
- 13.5k
13
votes
1
answer
3k
views
random walk and Brownian motion on Riemannian manifold
As we know, the random walk on $\mathbb{Z}/n$ will converge(in some sense) to the Brownian motion on $\mathbb{R}$ when $n\to\infty$. I would like to know is there some higher dimensional analogy ...
- 1,111
2
votes
1
answer
213
views
What happens to small squares in Riemann mapping?
I have a square S, and I want to convert it to the unit disc D.
The Riemann mapping theorem says that I can do this with a conformal bijective map. But, any such mapping will cause some distortion.
...
- 3,549
7
votes
0
answers
656
views
Least area minimal hypersurface of $\mathbb C P^{n+1}$
After a few lectures on min-max for minimal hypersurfaces and isoperimetric problems, and seeing in several instances that the least area minimal hypersurface of the round sphere is an equator, I was ...
- 5,891
4
votes
0
answers
396
views
Averaging lengths and distances
A natural way in which Finsler metrics appear in Riemannian geometry is as averages of Riemannian metrics (e.g., the average of the arc-length elements
$\sqrt{dx^2 + dy^2}$ and $\sqrt{2dx^2 + 3dy^2}$ ...
- 13.5k
14
votes
1
answer
3k
views
How metric is Riemannian geometry
Let $(M, g)$ be a finite-dimensional Riemannian manifold. It is well-known, that the Riemannian metric induce a metric on the manifold by
$$d(x, y) = \text{inf} \int_a^b \| \dot\gamma(t) \| \, dt\,,$$...
- 5,824
1
vote
1
answer
227
views
choices of connection in prequantization
In the definition of pre-quantization of representation $f\to \hat{f}$, (here $\hat{f}$ is Hermitian operator)of $C^{\infty}(M)$ on $L^2(M,L,\mu)$ where $\mu$ is Hermitian form, suppose that there ...
user21574
5
votes
2
answers
3k
views
Van Vleck-Morette Determinant
There seems to be something curious about the so-called Van-Vleck-Morette determinant, as I cannot find any source that properly defines it in terms of expressions previously defined in that source ...
- 9,853
3
votes
2
answers
324
views
Ito Diffusions with low regularity?
I would like to have an Itô Diffusion
$$ X_t = \int_0^t b(s) \mathrm{d}s + \int_0^t \sigma(s) \mathrm{d}B_s.$$
where the (vector- and matrix-valued, respectively) functions $b$ and $\sigma$ have lower ...
- 9,853
3
votes
1
answer
1k
views
Proof of a theorem of Jean-Pierre Serre on geodesics of closed Riemannian manifolds
An oft-cited theorem of Serre states that there are infinitely many geodesics between any two points in a closed Riemannian manifold. Could someone please provide an intuitive sketch of the proof?
- 135
1
vote
1
answer
177
views
pre-symplectic and foliation and its trajectories
Let $(M,\omega)$, be pre-symplectic, then can we say, we have a foliation of $M$, with tangent spaces $ker\omega$.What can we say about its trajectories. ?
user21574
1
vote
1
answer
568
views
A question on asymptotically flat metrics
For $M$ a Riemannian manifold, with Riemannian metric $g$ and $x$ a point in M, what is the meaning of "$g$ on $M\backslash\{x\}$ has an 'asymptotically flat end at $x$'."? (See this paper on page 16, ...
- 21
11
votes
1
answer
1k
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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
6
votes
0
answers
269
views
Negative curvature in the middle of $R^{3}$
What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside?
Basically, I am asking for a ...
- 61
5
votes
1
answer
1k
views
Existence of Geodesics in continuous metrics
I learned that if we are given a $C^0$ Riemannian metric on a smooth manifold $M$, geodesics (i.e. length minimizing curves) are absolutely continuous, and if the metrics is $C^{0,\alpha}$, then the ...
- 9,853
3
votes
2
answers
1k
views
Computations with the distance function on a Riemannian manifold
Let $(M,g)$ be a complete Riemannian, connected, compact manifold (with or without boundary). Let $f(r)$ be a decreasing function of $r =$ geodesic distance. If $\Omega \subset M$, then
$$ \int_{\...
- 207
8
votes
0
answers
438
views
Gromov-Hausdorff and Lipschitz convergence of a non-collapsing sequence of manifolds with Ricci curvature bounded below
There is a theorem from Cheeger-Colding saying the following:
Let $n$ be an integer. If a sequence of $n$-dimensional Riemannian manifolds $(M_i,g_i)$ converges with respect to the Gromov-Hausdorff ...
1
vote
1
answer
206
views
Riemann isometry vs Euclidean bi-Lipschitz mapping
Assume that $\gamma$ is a rectifiable Jordan curve in the complex plane of length $2\pi$. Then there exists a Riemann isometry $f$ between $\gamma$ and the unit circle $T$. My question is, does this ...
- 303