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2 votes
Accepted

Is every geodesic constant-speed?

Upshot from the comments: A geodesic always has constant speed, but not every constant-speed curve is a geodesic. The real feature which distinguishes a geodesic and some other curve seems to be the ...
user358572's user avatar
1 vote
Accepted

Proof a curve is a geodesic on a sphere S

I'll still post a solution because it could be handy in terms of the surfaces more complicated than a sphere. The basic idea was to somehow define the tangent plane, or, rather, separate it from the ...
Egor Larionov's user avatar
1 vote

Generator of translations in a Riemannian manifold

The Taylor formula has nothing to do with metrics. The partial derivatives act as generators along a a mesh considered as a flat, cartesian lattice. Any function like e.g. a metric translates by the ...
Roland F's user avatar
  • 3,279
1 vote

Sheaves of sections of vector bundles

I assume here $E$ and $F$ are both holomorphic vector bundles. In general the map obtains by tensorizing sections \begin{align*} F_{U} : \mathcal{O}(U,E)\otimes_{\mathcal{O}(U)} \mathcal{O}(U,F) \to \...
Alex-Antoine Caron's user avatar
1 vote

Vector bundle construction theorem

Thanks to @snailman I organise the complete and readable proof of the theorem with hopefully all one need to show. First, we construct the bundle and then prove its uniqueness. The construction is as ...
Gao Minghao's user avatar
1 vote
Accepted

Cheeger-Gromov limit of scalings of a noncompact Riemannian manifold

Take the quotient $M$ of the hyperbolic (upper half) plane by the cyclic group generated by the translation $z\mapsto z+1$. Take points $p_k\in M$ which are projections of $k\sqrt{-1}$. Use $r_k=\sqrt{...
Moishe Kohan's user avatar
1 vote
Accepted

On writing integration on Riemannian submanifolds in terms of exponential map

The answer is no, and the reason appears if you write properly what it means that "$N$ is determined by the exponential map". In rigorous terms, you are asking the exponential map $\exp_x$ ...
Didier's user avatar
  • 19.7k
1 vote
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Relation between the derivative and the covariant derivative?

There's an unfortunate typo in this proof, besides the $-$ part that you mentioned, the equation \begin{equation} \langle \frac{Dv}{dt}, w \rangle + \langle \lambda N \wedge v, w \rangle = [\frac{Dv}{...
Arctic Char's user avatar
  • 16.3k
1 vote

Statistical Model and its Topology

I will try to give an intuitive picture of the situation. Building on Fisher's work, Rao discovered that, on suitable manifolds of parametrized probability distributions like Normal distributions, the ...
fmc2's user avatar
  • 1,342
1 vote

Geodesic question

The geodesics of the exponential connection for the model you consider may be explicitly computed to be: $$ \mathbf{p}(t)=\frac{\mathbf{p}_{0}\,\mathrm{e}^{t\mathbf{a}}}{N(\mathbf{p}_{0},\mathbf{a},t)}...
fmc2's user avatar
  • 1,342

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