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|In short: k(M) is the set of smooth fields of exterior k-forms on the tangent spaces |
of M. The wedge-product, as defined in Section 6.2.1, can be extended to the
various k(M). We form the direct sum of the (infinite-dimensional) vector spaces k(
|1300-K. A880895 2150 Steel industry craft determination program, no. 1301-K. |
A880896 2150 Steel industry craft determination program, no. 1302-K. A880897
2150 Steel industry craft determination program, no. 1304-K, form A. A891895 ...
|Suppose wo is the 1-form on R* which we used to define the winding number (|
see Section 7.2). Let u(r, 0) ... If T is a differential k-form on M, then pl"T (the lift of
T) is a differential k-form on U which descends to a differential k-form on U/u ".
|We now generalize the construction of 1-forms on a manifold to k-forms. After |
defining k-forms on a manifold, we show that locally they look no different from k-
forms on Rn. In parallel to the construction of the tangent and cotangent bundles
|The vector-valued or tensor-valued k-forms can be added among themselves, |
multiplied by real numbers, by functions and by l-forms in the obvious way. The
set of vector-valued k-forms will be denoted by 3 (M) & A*(M), while the set of the
|(More generally, if a. is a k-form and g is an h-form, then wo = (–1)"go.) In terms of |
these algebraic operations on forms, the operation of forming the pullback of a k-
form a in y1, y2, ..., ym under an affine map y: = X aux + bi (i = 1,2,..., m) j=1 is ...
|Definition 3.1.3 A cubical differential k-form on M, with values in W, is a law (0, |
which to any infinitesimal k-dimensional parallelepipedum P in M associates an
element (0(P) G W, subject to the normalization requirement that (0(P) I 0 ifthe ...
|Hyperbolic Differential Forms Let M I Hn/F be a space-form with quotient map 71' |
: H" I M. Definition: A 0-form on M is a function f : M I R. If k is a positive integer,
then a k-form on M is a function _ k w . M I ugUMA such that w(u) is in Ak(Tu(M)) ...
|vector spaces, as we did in the case of l-forms, we obtain the vector-bundles of k-|
forms. More explicitly we can set: Definition 2.5.4 A differential k-form 600') is a
smooth assignment: at“) 1 p 1_> mg') 6 Ak(TpM) (2.5.54) of an exterior k-form on
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