Recent content by user2010

I am confused with the norms and the covariant derivatives. I know that ##||A||^2 = A_{ij} A^{ij}= g_{ik} g_{jl} A^{kl} A^{ij}## for a (0-2) tensor. So if ##\nabla X## is ##\nabla_i X^j = \partial_i X^j + \Gamma^j_{il} X^l ##, is ## | \nabla X|^2 ## equal to ## (\nabla_i X^j) (\nabla_j X^i) ## ?

Let ##(M,g)## a manifold with a Levi-Civita connection ## \nabla ## and ##X## is a vector field. 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 +...