Let M be a differentiable manifold of class C?. All tensor fields discussed below are assumed to be of class C?. Let X be a vector field on M. If X vanishes at a point 0 ? M then X induces, in a natural way, an endomorphism aX of the tangent space Vo at 0. In fact if y ? V0 and Y is any vector field whose value at 0 is y, then define axy = [X, Y]0. It is not hard to see that [X, Y]0 does not depend on Y so long as the value of Y at 0 is y.
Vector Field Riemannian Manifold Mathematical Method Mathematical Society Differential Geometry
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