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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. 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 ...
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