All Questions
Tagged with reference-request riemannian-geometry
320 questions
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Reference request: Riemannian manifold of linear isometries from $\mathbb{C}^n$ into $\mathbb{C}^m$
Does anyone know a citeable reference which works out the properties (geodesics, geodesic distance, ect) of the Riemannian manifold of linear isometries from $\mathbb{C}^n$ into $\mathbb{C}^m$, $m>...
- 25.3k
28
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2
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5k
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The Origin of the Musical Isomorphisms
In Riemannian geometry, the "lowering indices" operator is denoted by $\flat:TM \to T^*M$ and the "raising indices" operator by $\sharp:T^*M \to TM$. These isomorphisms are ...
CommunityBot
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6
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0
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352
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How to generate a random (Weyl) curvature operator ?
Given a dimension $n$, the space of curvature operators is the space $S^2_B(\Lambda^2\mathbb{R}^n)$ of symmetric endomorphisms $R$ of $\Lambda^2\mathbb{R}^n$ which satisfy the first Bianchi identity :
...
- 4,123
7
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1
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502
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Fundamental groups of compact manifolds with non-negative Ricci curvature.
I would like to find an appropriate reference for the following statement:
Statement. Let $M$ be a compact Riemannian manifold with non-negative Ricci curvature.
Then $\pi_1(M)$ is virtually abelian.
...
- 1,785
7
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1
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Open problems about CMC hypersurfaces with symmetries?
Recently, Andrews and Li announced a complete classification of CMC ($H=const.$) tori in $S^3$, confirming a conjecture of Pinkall and Sterling. Their main result is that any such torus is ...
- 5,891
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1
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Helmholtz-Decomposition on compact Riemannian manifolds
For smooth domains $\Omega$ in $\mathbb{R}^n$ it is known that one can decompose vector fields in $L^p(\Omega)^n$, $1 < p <\infty $ into a "gradient"- and a "divergence-free"-part such that
$L^...
CommunityBot
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7
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0
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695
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Sasaki Metric of the Tangent Bundle over the Hyperbolic Plane
This is a reference request on what are surely well known facts.
Let $M$ be a compact hyperbolic surface and $S(M)$ its unit tangent bundle. It follows from facts about Möebius tranformations in the ...
- 3,540
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1
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315
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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 ...
- 31.2k
2
votes
1
answer
551
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Heisenberg group: research themes
I am currently studying the Heisenberg group from the Riemannian geometry point of view, particularly focusing on its Gromov boundary and more generally its metric properties.
I would like to know ...
CommunityBot
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13
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4
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2k
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Algebraic surfaces and their (intrinsic) geometry
Recently I began to consider algebraic surfaces, that is, the zero set of a polynomial in 3 (or more variables). My algebraic geometry background is poor, and I'm more used to differential and ...
- 148k
7
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2
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965
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Full isometry groups of Stiefel and Grassmann manifolds
Hi,
I'm looking for a reference for the full isometry groups of the
(i) complex Stiefel manifolds $U(m)/U(m-l)$, either for the Euclidean metric (i.e. identifying it with orthonormal $m \times l$-...
CommunityBot
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6
votes
1
answer
1k
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How the Jacobi metrics may be useful in mechanics with or without constraints?
A mechanical system $(Q,K,V)$ is specified by the configuration space $Q,$ the potential energy $V\in C^\infty(Q),$ and the kinetic energy $K=K_g$ given by a Riemannian metric $g$ on $Q.$
If $V{<}...
- 13.5k
3
votes
2
answers
623
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convergence theory -> lorentzian geometry
Does someone have examples of extensions of results from convergence theory for riemannian geometry to a lorentzian setting. (I am familiar with the work of M.T.Anderson and co. in CMC gauge, i would ...
- 26
4
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1
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829
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Is there more than one closed geodesic on $S^3$?
I know from two sources
that it is (or at least was) unknown whether there are infinitely
many geometrically distinct closed geodesics
for every Riemannian metric on $S^3$, the 3-sphere
(Weinberger, ...
- 1,207
4
votes
2
answers
1k
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Special Killing Vector Fields
Consider $(M^{n},g)$ to be a Riemannian manifold and suppose that $X$ is a smooth non-trivial Killing vector field on $M$. Away from the zeros of $X$ we have a natural distribution $D$ of $(n-1)$-...
- 4,306
2
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1
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1k
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Harmonic coordinates on Riemannian manifolds
I'm trying to read the paper of Jost and Karcher on the existence of harmonic coordinates on a ball whose size only depend on the injectivity radius and a two sided bound on the curvature.
...
- 39k
4
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2
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734
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Analyzing the solution to a second-order, non-linear ODE
Let $\psi : [0,\infty] \to \mathbb R$ be a strictly positive, continuously differentiable function, and consider the non-linear ODE $$\ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - ...
- 60.5k
10
votes
3
answers
2k
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What spaces have well known horofunctions?
Following Gromov, take a metric space $(X,d)$ and consider $C(X)/\mathbb{R}$ the set of continuous functions to $\mathbb{R}$ with the topology of uniform convergence on compact sets after taking the ...
- 17k
6
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1
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342
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Contracting a geodesic on a space of curvature less than 1
I would like to ask for a reference to the following statement (hopefully correct):
Let $M$ be a manifold of sectional curvature at most $1$ and let $\gamma$ be a closed geodesic.
Suppose that $\...
- 45k
9
votes
2
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7k
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Constant curvature manifolds
In two different books I found these two related statements.
The book by Jost defines a ``locally symmetric space" as one for which the curvature tensor is constant and which is geodesically complete....
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