4 votes

Christoffel symbol on $T^*M$

Leibniz rule is, for $X$ and $Y$ vector fields and $\alpha$ a $1$-form, $$ \nabla_X(\alpha(Y)) = (\nabla_X\alpha)(Y) + \alpha(\nabla_XY). $$ Applying to $\alpha = dx^k$, $X= \partial_i$ and $Y=\...
Didier's user avatar
  • 18k
3 votes

Relation of connections on $TM$ and $T^*M$

Somehow things seem to be slightly messed up in your question. Initially, one defines a connection as sending $\Gamma^\infty(M,E)$ to $\Gamma^\infty(M,E)\otimes_{C^\infty(M,\mathbb R)}\Omega^1(M)$ ...
Andreas Cap's user avatar
  • 20.1k
3 votes

Reference request: English version of "Über eine Ungleichung von Cheeger "

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Anton Steinfadt's user avatar
3 votes

2nd Bianchi Identity via normal coordinates

First of all, we need to know that $$R_{ijk}^{\phantom{ijk}\ell} = \partial_i\Gamma_{jk}^\ell - \partial_j\Gamma_{ik}^\ell + \Gamma\Gamma + \Gamma\Gamma.$$We will not bother with the placement of ...
Ivo Terek's user avatar
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1 vote

Comprehension of the definition of 2nd partial derivatives on Riemannian manifolds.

For the second question. If we apply $e_i$ to the first equation we obtain, for any smooth function $\varphi$ $$ \varphi \circ f_i= e_i (\varphi \circ f) $$ If now $f_{ij} \omega^j= df_i$ we have $$ \...
Marco's user avatar
  • 815
1 vote

Is there analogue of four vertex theorem for Riemannian 2-sphere $(S^2,g)$?

One way of showing that the Gaussian curvature function on the 2-sphere may have as few as $2$ critical points is to use Gluck's result from Gluck, Herman. The generalized Minkowski problem in ...
Mikhail Katz's user avatar
  • 40.1k

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