ABSTRACT
A manifold is a topological space which locally looks like ℝ n . Calculus on a manifold is assured by the existence of smooth coordinate systems. A manifold may carry a further structure if it is endowed with a metric tensor, which is a natural generalization of the inner product between two vectors in ℝ n to an arbitrary manifold. With this new structure, we define an inner product between two vectors in a tangent space T p M. We may also compare a vector at a point p ∈ M with another vector at a different point p′ ∈ M with the help of the ‘connection’.
