Introduction to Riemannian Manifolds

Abstract

Let M (resp. N) be a connected. smooth (= C ^x) n-dimensional manifold without boundary. We denote by C ^x (M) the ring of smooth real valued functions on M and by x(M) the Lie-algebra of all smooth vector fields on M. Recall that X ∈ x(M) is a smooth map

$$X:M \to TM = \mathop U\limits_{x \in M} {T_x}M$$

such that X (x) = X _x , ∈ T _x M (= the tangent space of Mat x) for each x ∈ M. T _x M may be characterized as the space of all derivations of the algebra of smooth real valued functions defined on neighborhoods of x. Note that a vector field X and a function f ∈ C ^x (M) give rise to a new function X (f) ∈ C ^x (M) defined by

$$X(f)(x) = {X_x}(f).x \in M$$

.