Tagged Questions

2
votes
1answer
43 views

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 $t_f$ …
2
votes
2answers
205 views

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 o …
7
votes
4answers
558 views

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 …
5
votes
0answers
136 views

Prescribing Gaussian curvature

Let $K(r)$ be the piecewise function                        &nbs …
2
votes
1answer
120 views

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 i …
0
votes
2answers
212 views

Are all Riemannian metrics induced by Euclidean metrics? [Nash Embedding Theorem]

Let $M$ be a smooth manifold. We can get a Riemannian metric on $M$ by at least two methods: first by partitions of unity and second by the Whitney embedding theorem: we can embed …
1
vote
2answers
102 views

Structure constants to Christoffel Symbols

What is meant when one says that one has chosen a basis of fields on the manifold with ``anolonomy"? I get the feeling that it is a choice of basis with non-trivial structure cons …
3
votes
4answers
258 views

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 …
2
votes
4answers
173 views

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 gi …
8
votes
7answers
707 views

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 somewh …
2
votes
0answers
152 views

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 e …
3
votes
2answers
150 views

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 alo …
2
votes
4answers
234 views

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-inv …
4
votes
2answers
287 views

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 \p …
1
vote
0answers
125 views

Surface area of a hyperbolic paraboloid patch bounded by a skew quadrilateral

Given four non-planar 3D points, these points define a skew quadrilateral ABCD. Example: A = (0,0,0); B = (0,3,0); C = (3,3,2); D = (3,0,-1); The bilinear interpolation of …

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