# Tagged Questions

A branch of differential geometry dealing with Riemannian manifolds. *Riemannian manifolds* are smooth manifolds with an inner product smoothly attached to the tangent space of each point. Usually, Riemannian geometry focuses on the notions of distance, curvature, and shape. Consider using this tag ...

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### Metric on an open subset of $\mathbb{R}^d$ and Christoffel symbol of the second kind

I need to prove the following identity. $\Omega \subset \mathbb{R}^d$ is open and $g$ is a metric field on $\Omega$. Further $\Gamma_{\,kl}^j$ denotes the Christoffel symbol of second kind. $g^{ij}$ ...
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### The space of Riemannian metrics on a given manifold.

For a finite-dimensional smooth (Hausdorff, second-countable) manifold $M$, consider the set $$\mathcal{Met}(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$ I'd like to know about the typical ...
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### Existence of a Riemannian metric inducing a given distance.

Let $M$ be a smooth, finite-dimensional manifold. Suppose $M$ is also a metric space, with a given distance function $d: M \times M \rightarrow \mathbb{R}_{+}$, which is compatible with the original (...
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### Need help finding a good book on Riemann Geometry

I want to learn more about calculus on manifolds and Riemann Geometry. I have been reading the book Geometry, Topology and Physics by Nakahara. But I find that it is difficult to read due to the lack ...
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### Check Riemannian manifold's isometry to $\Bbb{R}^n$

Let $\mathcal{M}$ be the convex cone of symmetric positive definite $n\times n$ real matrices. $\mathcal{M}$ is an $\frac{n(n+1)}{2}$-dimenasional Riemannian manifold. Could you help me proving (or ...
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### The Riemannian Distance function does not change if we use smooth paths?

The Riemann distance function $d(p,q)$ is usually defined as the infimum of the lengths of all piecewise smooth paths between $p$ and $q$. Does it change if we take the infimum only over smooth paths?...
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### On a definition of manifold

In the book Mathematical Masterpiece, on page 160, the authors wrote that A manifold, in Riemann's words, is a continuous transition of an instance I know a manifold is something glued by ...
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### Geodesics on the product of manifolds

Given two Riemannian manifolds $(M, g_1)$ and $(N, g_2)$, and geodesic curves $\gamma(t)$ in $M$ and $\chi(t)$ in $N$. Is the curve $\Gamma(t) = (\gamma(t),\chi(t))$ a geodesic in the product manifold ...
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I am considering the unit sphere (but an extension to one of radius $r$ would be appreciated) centered at the origin. Any coordinate system will do, though the standard angular one (with 1 radial and $... 3answers 5k views ### Is there a good way to compute Christoffel Symbols Lets say you have a Riemannian Manifold$(M,g)$, and you have some given chart where$g = g_{ij} dx_i dx_j$and you wish to compute the Christoffel symbols for the Riemannian connection in this chart. ... 1answer 2k views ### Isometries preserve geodesics Let$f$be an isometry (i.e a diffeomorphism which preserves the Riemannian metrics) between Riemannian manifolds$(M,g)$and$(N,h).$One can argue that$f$also preserves the induced metrics$d_1, ...
Im studying DoCarmo's book Riemannian Geometry, the first problem of the chapter 5 (Jacobi Fields) states that If $(M,g)$ is a riemannian manifold with sectional curvature identically zero, show that ...