Let ##(M,g)## a manifold with a Levi-Civita connection ## \nabla ## and ##X## is a vector field.(adsbygoogle = window.adsbygoogle || []).push({});

What is the formula of ## | \nabla X|^2 ## in coordinates-form?

I know that ##|X|^2= g(X,X)## is equivalent to ## X^2= g_{ij} X^iX^j## and ##\nabla X## to ##\nabla_i X^j = \partial_i X^j + \Gamma^j_{il} X^k ## but I can't use these to ## | \nabla X|^2 ##.

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# I Norm of the Laplacian

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