Norm of Laplacian Let: Formula for | ∇X|² in Coordinates

[user2010]

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 $\nabla_i X^j = \partial_i X^j + \Gamma^j_{il} X^k$ but I can't use these to $| \nabla X|^2$.

 

[Orodruin]

What do you mean by $|\nabla X|^2$? The object $\nabla X$ is a type (1,1) tensor and if you want to compute its norm you need to define it.

 

[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 $(\nabla_i X^j) (\nabla_j X^i)$ ?