All Questions
Tagged with dg.differential-geometry riemannian-geometry
1,985 questions
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What are the computationally useful ways of thinking about Killing fields?
One definition of the Killing field is as those vector fields along which the Lie Derivative of the metric vanishes. But for very many calculation purposes the useful way to think of them when dealing ...
- 3,541
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votes
3
answers
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Curvature of a Lie group
Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold [Edit: provided we fix a metric, for example an invariant or bi-invariant one] has some scalar ...
- 251
2
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1
answer
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Frobenius Theorem
Say a manifold M has 3 vector fields S,T and R whose Lie brackets satisfy the equations $[S,T]=R$, $[R,S]=T$ and $[T,R]=S$
Then I suppose the following properties hold for M,
There exists a metric ...
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votes
10
answers
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Some questions about scalar curvature
Recall that the scalar curvature of a Riemannian manifold is given by the trace of the Ricci curvature tensor. I will now summarize everything that I know about scalar curvature in three sentences:
...
- 29.2k
-1
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1
answer
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Harmonic maps in the cotangent bundle
$M$ is a Riemannian manifold with metric $g$ and we have a map $F: M \to T^{\*}M$ with $F(p)=(p,f(p))$ with a 1-form $f$. On $T^{*}M$ we use the Sasaki-metric.
How can I prove or it is wrong?:
$F$ ...
3
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1
answer
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Determinant of a metric?
In a paper that I am reading, the author is weighting edges in a graph using
$$w_k \propto \det(D(p))$$
where $D(p)$ is the metric tensor (which if I understand correctly is a space-varying metric?). ...
- 171
5
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0
answers
2k
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Relationship between geodesics and curvature lines on surfaces?
I'm trying to understand the relationship between geodesics and lines of principal curvature (to keep things simple, let's say Riemannian 2-manifolds embedded in $\mathbb{R}^3$). In my reading, I ...
- 3,059
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2
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Diameter of a circle in an embedded Riemannian manifold
This question was inspired by an answer to the "Magic trick based on deep mathematics" question. I wanted to post it as a comment, but I ran out of characters! I'm sure there must be a collection of ...
- 2,284
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1
answer
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Global description of the Levi-Civita connection
I'm interested in finding a global (coordinate-free) description of the Levi-Civita connection on a (possibly infinite-dimensional) Riemannian manifold X.
I'm not looking for a description of this ...
- 8,803
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2
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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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4
votes
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answers
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Flat Riemanniann manifolds
Are there Riemanniann manifolds with zero curvature other than open subsets of $\mathbb{R}^n \times \mathbb{T}^m$, where $\mathbb{T}^m$ is an $m$ dimensional torus and $m,n\geq 0$ ?
Does taking ...
- 23.3k
9
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Non-Kahler manifolds where the different Laplacians are compatible
On a Kahler manifold, the different Laplacians are compatible: $\Delta_d=2\Delta_{\bar{\partial}}=2\Delta_{\partial}$.
Are there non-Kahler Hermitian manifolds where the above identity holds?
user2529
40
votes
0
answers
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Minimal volume of 4-manifolds
This question came up in a talk of Dieter Kotschick yesterday. The minimal volume of a manifold is the infimum of volumes of Riemannian metrics on the manifold with sectional curvatures bounded in ...
- 68.9k
41
votes
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Introductory text on Riemannian geometry
I have studied differential geometry, and am looking for basic introductory texts on Riemannian geometry. My target is eventually Kähler geometry, but certain topics like geodesics, curvature, ...
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4
answers
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Does every smooth manifold of infinite topological type admit a complete Riemannian metric?
To elaborate a bit, I should say that the question of the existence of a complete metric is only of interest in the case of manifolds of infinite topological type; if a manifold is compact, any metric ...
- 1,665
7
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4
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Existence of Fermi coordinates on a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold, $p$ a point on the manifold and $v \in T_p M$. Let $\gamma$ be the geodesic starting at $p$ in the direction $v$. There exists a time $t_f$ such that there ...
- 8,512
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Equivalent singular chains and differential forms, as functionals on forms, on compact Riemannian manifolds
On a compact Riemannian oriented manifold $M$,for each singular $k$-chain $\sigma$ (with real coefficients), $\sigma$ induces a linear functional on the $\mathbb{R}$-vector space of differential k-...
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5
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1
answer
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Orthogonal complements in Hilbert bundles
It's a standard fact that for a finite-dimensional vector bundle with an inner product, the othogonal complement of any subbundle is itself a locally trivial vector bundle.
What is known about the ...
- 8,803
24
votes
5
answers
6k
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Curvature and Parallel Transport
Here is an updated formulation of the question, which is more precise and I think completely correct:
Suppose $M$ is a Riemannian manifold. Pick a point $p$ in $M$ and let $U$ be a neighborhood of ...
- 29.2k
5
votes
4
answers
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Testing for Riemannian isometry
In most physics situations one gets the metric as a positive definite symmetric matrix in some chosen local coordinate system.
Now if on the same space one has two such metrics given as matrices then ...
- 3,541
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votes
1
answer
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Surjectivity of the normal exponential map
Given an isometric (in the Riemannian way) immersion $f:N\rightarrow M$ between complete, smooth riemannian manifolds, are there conditions on $M$, $N$, $f$, such that the normal exponential map $\...
- 1,452
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votes
3
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Changing coordinates so that one Riemannian metric matches another, up to second derivatives
Let $g$ and $g'$ be two $C^2$-smooth Riemannian metrics defined on neighborhoods $U$ and $U'$ of $0$ in $\mathbb R^2$, respectively. Suppose furthermore that the scalar curvature at the origin is $K$ ...
- 8,512
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3
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Jacobi fields on a "bump surface"
Consider a "bump surface" which looks like the following:
Such a surface is rotationally symmetric, $C^2$-smooth, has positive curvature in the middle and negative curvature along the ring (the ...
- 8,512
6
votes
4
answers
3k
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Killing fields on homogeneous spaces
Let $G$ be a compact lie group and $H$ a closed subgroup and hence think of $G/H$ as a homogeneous space.
Then how are the Killing fields on $G/H$ the projection of the right-invariant vector fields ...
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7
votes
1
answer
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The orthodrome of n-spheres.
I am a Computer Science undergraduate who does a lot of other tinkering in his free time. Right now, I'm tinkering with n-spheres. Specifically, I'm looking at the distances between a collection of ...
- 73
15
votes
8
answers
6k
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Riemannian Geometry
I come from a background of having done undergraduate and graduate courses in General Relativity and elementary course in riemannian geometry.
Jurgen Jost's book does give somewhat of an argument ...
- 3,541
7
votes
2
answers
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Why these particular numerical factors in the definition of Gaussian curvature?
Wikipedia tells me that:
Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane:
$K = \lim_{r \rightarrow 0} (2 \pi r - \mbox{C}(r)) \...
- 3,217
5
votes
5
answers
3k
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Tetrad postulate: Implies or results from the metricity of the connection?
Hi,
I see that the tetrad postulate:
$\nabla_{\mu}e_{\nu}^{I}=\partial_{\mu}e_{\nu}^{I}-\Gamma_{\mu\nu}^{\rho}e_{\rho}^{I}+\omega_{\mu J}^{I}e_{\nu}^{J}=0$
Can be merely derived from writing a ...
- 73
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votes
4
answers
3k
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When is a Riemannian manifold an open subset of a complete one?
Under what conditions can a Riemannian manifold be embedded isometrically as a submanifold of a complete one of the same dimension? There should some kinds of necessary conditions. For instance, ...
- 25.6k
14
votes
4
answers
6k
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When is a Riemannian metric equivalent to the flat metric on $\mathbb R^n$?
I'm looking for an easily-checked, local condition on an $n$-dimensional Riemannian manifold to determine whether small neighborhoods are isometric to neighborhoods in $\mathbb R^n$. For example, for ...
- 54.6k
5
votes
1
answer
564
views
Lower bound on volume of minimal hypersurface contained in a unit ball with curvature bounds
I was just wondering, if I have a geodesic ball of radius one in a manifold M whose sectional curvature lies between -epsilon and epsilon for epsilon small, and the injectivity radius of my manifold ...
- 51
24
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5
answers
2k
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Point singularity of a Riemannian manifold with bounded curvature
Suppose you have an incomplete Riemannian manifold with bounded sectional curvature such that its completion as a metric space is the manifold plus one additional point. Does the Riemannian manifold ...
- 27.5k
32
votes
4
answers
4k
views
Largest hyperbolic disk embeddable in Euclidean 3-space?
Hilbert proved that there's no complete regular ($C^k$ for sufficiently large $k$) isometric embedding of the hyperbolic plane into $\mathbb{R}^3$. On the other hand, the pseudosphere is locally ...
- 13.6k
6
votes
2
answers
2k
views
Eigenvalues of Laplacian
What's the most natural way to establish the asymptotics of $\Delta$ on a compact Riemannian manifold $M$ of dimension $N$? The asymptotics should be
$$ \#\{v < A^2\} = \mathrm{const}\ast\mathrm{...
- 15.1k