# 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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### Show that $-\langle\nabla|\text{Rm}|^2,\nabla|\nabla \text{Rm}|^2\rangle\le4|\text{Rm}||\nabla \text{Rm}|^2|\nabla^2 \text{Rm}|$

Here $\text{Rm}$ is the curvature tensor. When I try to compute $$-\langle\nabla|\text{Rm}|^2,\nabla|\nabla \text{Rm}|^2\rangle\le4|\text{Rm}||\nabla \text{Rm}|^2|\nabla^2 \text{Rm}|,$$ I compute ...
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### Compute of a inequality about cutoff function

How to compute the inequality with red line in the below picture ? It seems to integrate the inequality 1 , but I don't know why there is $e^{CMt}$ and where the $\int_0^t\varphi|\nabla^2 \varphi|$ ? ...
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### Symmetry of Kahler metric on based loop group

The based loop group, $\Omega G$, is known to admit a Kaehler metric, given as $$g(X,Y)=2\sum_{k>0}k\textrm{Tr}(X_{-k}Y_k),$$ this is given in page 150 of Segal and ...
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### How to define the smooth of vector field on Riemannian manifold? [closed]

$(M,g)$ is a Riemannian manifold, if $X$ is a vector field on $M$, I think for differential points $p,q\in M$, $X_p$ and $X_q$ are belong to differential space $T_pM$ and $T_qM$, I can't image how to ...
### Image of a path $\gamma$ is totally geodesic $\Rightarrow \gamma$ is a reparametrization of a geodesic?
Let $M$ be a Riemannian manifold. Assume $\gamma$ is a path in $M$ , such that it's image is a totally geodesic submanifold of $M$. I am trying to prove (the seemingly trivial result) that $\gamma$ ...