Understanding the Contravariant Derivative: A Tangential and Normal Perspective

[bchui]

So much has been talking about covariant derivative. Anyone knows about contravariant derivative? What is the precise definition and would that give rise to different \Gamma^{k}_{i,j} and other concepts?

 

[robphy]

A "contravariant derivative operator" would probably be defined by \nabla^a=g^{ab}\nabla_b, where \nabla_b is a torsion-free derivative operator that is compatible (\nabla_a g_{bc}=0) with a nondegenerate metric g_{ab}.

 

[dextercioby]

The connection needn't be torsion free, but metric compatibility is essential.

Daniel.

 

[bchui]

Let (M,{\cal T}) be a sub-manifold of a Riemannian manifold (N,{\cal R}) with metric tensor g, If we decompose the tangent space at the point p\in M\subseteq N and accordingly decompose the tangent bundle T_pN=T_pM\circleplus {\tilde T}_pM into tangential to M and normal to M, could we say that the "converiant derivative" is the "tangential component" of the given connection \nabla_X: {\cal X}(N)\mapsto {\cal X}(N) while the "contravariant derivative" is the "normal component" of \nabla_X ?
I mean the "convariant derivative along the vector fileld X" is the projection of \nabla_X onto the tangent space of the submanifold M, while the "contravariant derivative along the vector field X" is the projection of X onto the normal space of the submanifold M in N
I would like to check if the above saying is correct