# 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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### What is the geometric interpretation of the Koszul formula?

I saw this simple form of Koszul formula on a book: $$2\ g(\nabla_XY,Z) = \mathcal{L}_Yg(X,Z) + (d\theta_Y)(X,Z)$$ where $\theta_Y$ is the one-form $g(Y,\cdot)$. It is equivalent to the more ...
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### Lower bound on convexity radius in terms of injectivity radius (without using curvature)

Let $M$ be a complete Riemannian manifold, and let $C$ be a subset of $M$. We will say $C$ is convex if for any points $p,q \in C$, there exists a unique normal minimal geodesic $\gamma$ joining $p$ ...
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### Differential Geometry - Distributions mutually orthogonal, span the tangent space, parallel imply manifold splits locally as product manifold

I'm stuck on a portion of Exercise 21, Chapter 2 in Petersen's Riemannian geometry text. Fix a Riemannian manifold $(M,g).$ Suppose that I have two distributions $D^1$ and $D^2$ defined on $M.$ ...
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### Geodesics on Lorentzian (2n-1)-Spheres

I know that if we endow $S^{n}$ with the round Riemannian metric, we will be able to join the North pole and the South pole by an unlimited number of geodesics, in particular the meridians, and indeed ...
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### Characterization of locally conformally flat manifolds with Frobenius theorem

In Hamilton's Ricci Flow (by Chow, Lu, Ni, pp. 29-31, see here) they show that a Riemannian manifold $(M^n,g)$ is locally conformally flat iff the Weyl tensor vanishes (when $n\ge 4$) and iff the ...
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### Invariance of ball size under translation on the pseudosphere

On the pseudosphere, do the volume of metric balls and area of metric spheres change with the centerpoint? (I'm asking because they don't on the sphere.)
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### Manifolds with volume forms on every submanifold

If we equip a manifold with an inner product (i.e. we have a Riemannian Manifold) then we get a canonical volume form on that manifold (please mentally insert the prefix "pseudo" into my question ...
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### Connectedness and Hopf-Rinow Theorem

Does the Hopf-Rinow theorem hold if the Riemannian manifold is not necessarily connected? $\\$ $\bf{Motivation \ and \ Minor \ Details \ About \ Question:}$ I am reading a non-standard book which ...
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### How to prove that $\forall$ Riemannian manifold $M$, $T\in C^{1}(M)$ and $x,y\in M$ close enough: $d(Ty,exp_{Tx}(D_xTexp^{-1}_xy))\leq d(x,y)$?

How to prove that $\forall$ Riemannian manifold $M$, $T\in C^{1}(M)$ and $x,y\in M$ close enough: $$d(Ty,\exp_{Tx}(D_xT\ \exp^{-1}_xy))\leq d(x,y)$$ My attempts so far were only able to show the ...
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### An application of Nash's embedding theorem to manifolds with fixed volume form

I have a smooth (possibly compact, or closed, or oriented, or more than one of the previous) $n$-manifold $M$ together with a fixed volume form $\rho\in\Omega^n(M)$. Can $M$ be embedded into some ...
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