All Questions
Tagged with reference-request riemannian-geometry
320 questions
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A reference for Poincaré's type inequality for vector fields
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\TM}{T\mathcal{M}}$
$\newcommand{\Ric}{\operatorname{Ric}}$
$\newcommand{\Volg}{\operatorname{Vol}_g}$
I would like to find a reference for the following ...
- 6,741
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Gromov-Hausdorff relative compactness without curvature restrictions
A famous theorem of Gromov says that the set of compact Riemannian manifolds with $Ric \geq c$ and $\text{diam} \leq D$ is relatively compact in the Gromov-Hausdorff metric. Chapter 10 of the book by ...
- 1,407
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133
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Reference for example of gradient steady Ricci solitons
Recently I read a paper about Ricci solitons. I quote a paragraph of it here:
In dimension three, the classification of complete gradient steady Ricci solitons is still open. Known examples are ...
- 4,195
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Proof of Berger in "Sur les variétés d’Einstein compactes"
I would appreciate any reference that contains either a translation or proof of the following interesting observation of Berger (Sur les variétés d’Einstein compactes, M. Berger paper (in French)).
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- 4,195
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151
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Asymptotic estimation of numbers of unlabeled graphs whose degrees of vertices are bounded
It is known(Enumeration of graphs with a given and bounded degree sequence) that there is no a closed form formula for number of (labeled) graphs with bound on degree of vertices. Thus what I want to ...
- 8,071
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82
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Scattering in (pseudo-)Riemannian spaces
I will ask my question in a broad way, leaving a lot of freedom for answers.
Suppose that we have a (pseudo-)Riemannian space $(M,g)$ and we fix some ball-like domain $B \subset M$. Suppose you are ...
- 1,461
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224
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Characterization of the Riemann curvature tensor [duplicate]
Let $(M^n,g)$ be a Riemannian manifold, $a\in M$ be a fixed point. It it well known that there exists a coordinate system near $a$ (e.g. the normal one) such that
$$g_{ij}(x)=\delta_{ij}+O(|x|^2).$$
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- 21.8k
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463
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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
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339
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Polarisation in a neighbourhood of a Lagrangian submanifold
Let $(X, \omega)$ be a symplectic manifold of dimension $2n$ and $\omega$ is an exact symplectic form i.e. $\omega = -d\alpha$. Let furthermore $M \subset X$ be a compact Lagrangian submanifold such ...
- 339
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Explicit computation of the vertical and horizontal vector bundles
Given a closed Riemannian manifold $(X,g)$ and let $p\colon TX\to X$ be the usual projection, the paper I'm reading asserts that the Levi-Civita connection induces a splitting $T(TX)= H(TX)\oplus V(TX)...
- 247
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Is this a manifold of bounded geometry?
Let $M$ be a complete non-compact manifold (possibly with boundary). Let $E$ be an open proper connected non-precompact subset of $M$ with smooth topological boundary, so that $\overline{E}$ is a non-...
- 459
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108
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Intersection Grassmanian planes
I am reading a paper that used Grassmanian planes properties. In particular, they studied the intersection of Grassmanian planes; they check the intersection Grassmanian of $n-k$-planes and ...
- 1,043
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314
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G-structures and complete riemannian manifolds
what are possible fundamental and introductory texts about G-structures ?
and where i can find the proof of this proposition:
if G(group) acts properly discontinuously on a space X , then G is a ...
- 165
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Request for resources on directional derivative of the Riemannian distance function, and Berger's lemma about geodesics realizing the diameter
I've been recently interested in directional derivatives of the Riemannian distance function, and I came across this question, and its answer by Sergei Ivanov, where he stated an important result: (I ...
- 1,467
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425
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Compact connected Riemannian manifolds are Ahlfors regular metric space
Let $(M,g)$ be a compact connected $n$-dimensional Riemannian manifold; let $(X,d)$ denote its associated metric (length) space. A comment on the original formulation of this post mentioned that $(X,...
- 5,405
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References for local distance approximation over Riemannian manifolds [duplicate]
Over a complete Riemannian manifold $(M,g)$, in a neighborhood of $p \in M$, the local distance can be approximated as follows: $\forall v,u \text{ unit vectors in } T_pM, \text{ and small } s, t$
$$ ...
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94
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Symmetry group for (noncommutative) manifold from spectral triple
(This post is cross-post in Mathematics Stack Exchanges https://math.stackexchange.com/questions/3992766/symmetry-group-for-noncommutative-manifold)
Is there any notion of symmetry group arise for ...
- 523
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126
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mean curvature for codimension $>1$?
The mean curvature of a hypersurface in a Riemannian manifold is defined to be the trace of the second fundamental form. I was curious, does the notion of mean curvature generalise to higher ...
- 3,625
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69
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optimal frame for a Riemann metric
In his excellent book, Harmonic maps, conservation laws and moving frames, Helein proves the existence of conformal coordinates on a surface, looking for an optimal frame. Let $\mathbb{D}$ equipped ...
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Obtaining Hessian of the embedding from an induced metric
Consider a hypersurface (not necessarily compact) smoothly embedded into $\mathbb{R}^n$ such that the Hessian is a positive definite bilinear form. Due to positivenes, Hessian can be taken as a metric ...
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