# Tag Info

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 ...
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 ...
• 1,218
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 ...
• 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 \...
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 ...
• 123
1 vote
Accepted

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 $r_k=\sqrt{... • 102k 1 vote Accepted ### On writing integration on Riemannian submanifolds in terms of exponential map The answer is no, and the reason appears if you write properly what it means that "$N$is determined by the exponential map". In rigorous terms, you are asking the exponential map$\exp_x$... • 19.7k 1 vote Accepted ### Relation between the derivative and the covariant derivative? There's an unfortunate typo in this proof, besides the$-$part that you mentioned, the equation \langle \frac{Dv}{dt}, w \rangle + \langle \lambda N \wedge v, w \rangle = [\frac{Dv}{... • 16.3k 1 vote ### Statistical Model and its Topology I will try to give an intuitive picture of the situation. Building on Fisher's work, Rao discovered that, on suitable manifolds of parametrized probability distributions like Normal distributions, the ... • 1,342 1 vote ### Geodesic question The geodesics of the exponential connection for the model you consider may be explicitly computed to be:$\$ \mathbf{p}(t)=\frac{\mathbf{p}_{0}\,\mathrm{e}^{t\mathbf{a}}}{N(\mathbf{p}_{0},\mathbf{a},t)}...
• 1,342

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