lect01 - SYMPLECTIC GEOMETRY, LECTURE 1 Prof. Denis Auroux...

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Unformatted text preview: SYMPLECTIC GEOMETRY, LECTURE 1 Prof. Denis Auroux 1. Differential forms Given M a smooth manifold, one has two natural bundles: the tangent bundle T M = = v i x i } and { v the cotangent bundle T M = { = i dx i } . Under C maps, tangent vectors pushforward: (1) f : M N = f ( v ) = df ( v ) T f ( v ) N Similarly, differential forms pull back: f ( ) = df T p M . Definition 1. A differential p-form is a section of p T M . We denote the set of such sections as p (2) p ( M ) = p ( M, R ) = C ( T M ) Recall that, for E a vector space, E = E/ { e i e j + e j e i = 0 } . Furthermore, E has a basis e i 1 e i p , i 1 < < i p . In coordinates, a p-form is locally (3) = i 1 , ,i p dx i 1 dx i p i 1 < <i p where the i 1 , are C functions. (Under coordinate changes, = f i ( y 1 , . . . , y n ), one replaces dx i by ,i p x i f i df i = j y j dy j .) Definition 2. The exterior differential is the map d : p p +1 which maps: f For f a function, df = x i dx i . d ( fdx i 1 dx i p ) = df dx i 1 dx i p . d is obtained by extending R-linearly to all of p . Note that d satisfies d ( f ) = fd + df . The exterior derivative has the following properties: d ( ) = ( d ) + ( 1) deg d . In coordinates, (4) d (( fdx i 1 dx i p ) ( gdx j 1 dx j q )) = ( fdg + gdf ) dx i 1 dx i p dx j 1 dx j q d 2...
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This note was uploaded on 03/08/2010 for the course MATHEMATIC 570 taught by Professor Denisauroux during the Spring '10 term at Caltech.

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lect01 - SYMPLECTIC GEOMETRY, LECTURE 1 Prof. Denis Auroux...

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