4
votes
Accepted
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=\...
- 18k
3
votes
Accepted
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 $\Gamma^\infty(M,E)\otimes_{C^\infty(M,\mathbb R)}\Omega^1(M)$ ...
- 20.1k
3
votes
Accepted
Reference request: English version of "Über eine Ungleichung von Cheeger "
Did you notice this article? It might be relevant.
- 181
3
votes
Accepted
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 ...
- 76.4k
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
$$
\...
- 815
1
vote
Accepted
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 ...
- 40.1k
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