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03 - algebra of vector fields

# 03 - algebra of vector fields - INTRODUCTION TO MANIFOLDS...

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INTRODUCTION TO MANIFOLDS — III Algebra of vector fields. Lie derivative ( s ) . 1. Notations. The space of all C -smooth vector fields on a manifold M is denoted by X ( M ). If v X ( M ) is a vector field, then v ( x ) T x M R n is its value at a point x M . The flow of a vector field v is denoted by v t : t R v t : M M is a smooth map (automorphism) of M taking a point x M into the point v t ( x ) M which is the t -endpoint of an integral trajectory for the field v , starting at the point x . Problem 1. Prove that the flow maps for a field v on a compact manifold M form a one-parameter group: t, s R v t + s = v t v s = v s v t , and all v t are diffeomorphisms of M . Problem 2. What means the formula d ds fl fl fl fl s =0 v s = v and is it true? 2. Star conventions. The space of all C -smooth functions is denoted by C ( M ). If F : M M is a smooth map (not necessary a diffeomorphism), then there appears a contravariant map F * : C ( M ) C ( M ) , F * : f 7→ F * f, F * ( x ) = f ( F ( x )) . If F : M N is a smooth map between two different manifolds, then F * : C ( N ) C ( M ) . Note that the direction of the arrows is reversed! Problem 3. Prove that C ( M ) is a commutative associative algebra over R with respect to pointwise addition, subtraction and multiplication of functions.

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03 - algebra of vector fields - INTRODUCTION TO MANIFOLDS...

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