# 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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### How much classical geometry must a geometer know?

From my reading Wikipedia, I understand there are several branches of classical geometry (if the ordering is off, or I'm missing a few things, let me know): Absolute Euclidean Non-Euclidean ...
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+50

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### No conjugate points on $S^1\times \Bbb R$

Lee claims in his book that $S^1\times \Bbb R$ (considered as a submanifold of $\Bbb R^3$) admits no conjugate points along any geodesic. I am struggling to make that rigorous. Being conjugate along ...
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### Condition for a one-parameter family of maps to be isometries

Given two smooth manifolds with Riemannian metric $(X,g)$ and $(Y,h)$ and a smooth map $f: X \to Y$ I understand that we define $\phi$ to be an isometry if $f^* g = h$. I thought I understood this ...
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### Algebraic topology & Riemannian geometry project idea?

I'm taking a first course on Riemannian geometry this semester. For a final project, I would like to do something that involves algebraic topology. However, the only results I know in algebraic ...
I'm studying from a textbook and came across a theorem as the following, which it calls the Beltrami Theorem: Beltrami Theorem: A Riemannian metric $g$ is projectively flat if and only if it is ...
For some reasons I need to show the following fact. Let $(M, g)$ be a Riemannian manifold. Let $U \subset M$ be an open set and $r: M \to \mathbb{R}$ a smooth distance function. Let us assume ...