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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Why $\nabla _{\dot \gamma (t)}Y_t=\dot x^i \frac{\mathrm d a^j(t)}{\mathrm d t}\partial _j+\dot x^i a^j\nabla _{\partial _i}\partial _j$

Let $M$ a smooth manifold and $\nabla $ a connexion. Let $\gamma :[a,b]\longrightarrow M$ a $\mathcal C^\infty $ curvature. I recall that if $X,Y\in \Gamma(M)$, and $f,g\in \mathcal C^\infty (M)$, ...
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Distance function on complete Riemannian manifold.

Let $\left(M, g\right)$ be a complete Riemannian manifold. Let us fix a point $p \in M$ and consider the distance function $$ r(x) := \operatorname{dist}(x, p). $$ I would like to characterize the ...