I have a riemannian 2-manifold (Let's say a sphere) and a tensor of order 1 (momentum), defined on the manifold. I want to integrate the tensor on a local neighborhood around a point p. More precisely, I want to know if the linear momentum is conserved on that part of the manifold using only intrinsic quantities. In a book, they say that they define a parallel vector field on that neighborhood so they can do a dot product between the two tensors. But I'm really not convinced of this method because there no "absolute" parallel field on surface and each field will give different result. How can I do that?