# 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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### Why Differential Forms on Riemann surfaces?

I am working with Rick Miranda's "Algebraic Curves and Riemann Surfaces". Right now I am in chapter four "Integration on Riemann Surfaces" and struggle with it a lot!:( It starts with the definition ...
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### Geodesic flow generated by Riemannian distance function

This is an exercise in AC da Silva's Lectures onn Symplectic Geometry; I am having trouble showing the following. $(X,g)$ is a geodesically complete manifold, and $f: X \times X \to \mathbb{R}$ is ...
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### Proof of iff condition of harmonic map

How to compute the equation above the red line in the picture below ? Below picture is from the Harmonic maps and their heat flows. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
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### how to define complete pseudo-riemannian manifold

In riemannin geometry, we define distance function by minimizing the length of curves. However we have nondefinite metric on psedu-riemannian manifold, so we cannot define a length of a curve as ...
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### Flag Curvature in Finsler Geometry

Does anyone know what is the flag curvature in Finsler geometry? I looked for this definition, but I don't find any answer.
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### About transformations of the metric: should we use the old or the new one to raise/lower indices?

Let $(M,g)$ be a (Pseudo-)Riemannian manifold. If I perform a transformation on the metric, getting a new metric $\tilde{g}$, which metric should I use to raise and lower indices? As I understand, ...
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### Riemannian connection on Lie groups

Let $G$ be a Lie group with a bi-invariant metric. Then, the Riemannian connection is given by $\nabla_XY=\frac1 2 [X,Y]$ for all $X,Y\in \mathfrak g$. In the proof: Since $\langle X,Y\rangle$ is ...
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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 ...
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### Vanish christoffel symbols implies $g_{ij,k}=0$

Let $(M,g)$ be a pseudo-riemann manifold and $(U,\psi=(x^1,\ldots,x^n))$ a local chart around some point $p$ in $M$. It is easy to show that if $\partial g_{ij}/\partial x^k=0$ in $p$ for all $i,j,k$ ...
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### Problem to conceptualize $\frac{\partial }{\partial \theta}$ and $\frac{\partial }{\partial \varphi}$.
I have some little problem to give a conception to $\frac{\partial }{\partial \theta}$ and $\frac{\partial }{\partial \varphi}$ on manifold (like $\frac{\partial }{\partial x}$ as well). For example, ...