00000154.htm

20 DIFFERENTIAL FORMS 137

where TT is the canonical mapping onto the orbit space (12.10.6) and j is a homeo-
morphism onto a closed subset V_n of C^N. The canonical image of V_n — {0} in the pro-
jective space P_N~i(C) is analytically isomorphic to P_n_i(C).

(b) The complex manifold obtained by blowing up the point 0 in C^N (Section 16.11,
Problem 3) is analytically isomorphic to the canonical bundle L_N__lfC(l). If

is the canonical projection of this blowing-up (it is a local isomorphism everywhere
outside the fiber q~^l(Q)), show that q~*(V_n) is analytically isomorphic to the bundle
L_n_i,_c(2) over P_w_i(C) (Section 16.16, Problem 1).

20. DIFFERENTIAL FORMS

(16.20.1) Let M be a differential manifold. The dual T(M)* of the tangent
bundle T(M) is called the cotangent bundle of M. If F is the transition diffeo-
morphism between two charts on M, then, as we have seen, the transition
diifeomorphism between the associated fibered charts of T(M) is

(16.15.4.5). Hence the transition diffeomorphism between the associated
fibered charts of T(M)* is (x, h*)h+(F(*), 'DF(jc)-¹ • h*).

We shall write TJ(M) in place of T*J(T(M)) when there is no risk of con-
fusion; in particular, therefore, TJ(M) = T(M) and T?(M)=T(M)*. A
section of TJ(M) over a subset A of M is called a tensor field (or, by abuse of
language, a tensor) of type (p, q) over A. The set F(M, T£(M)) of tensor fields
of class C°° over M is denoted by «^~J(M) or ^_)R(M); it is a module over the
ring rf(M) = <$(M ; R) (also denoted by <^_R(M)j of real-valued C°° -functions
on M, and is &free module when T(M) is trivializable (16.15.8).

For each p ^ 1, a section over A of the bundle /\T(M)* of tangent p-
covectors is called a differential p-form on A (or simply a differential form

when p = 1). The set F(M, y\T(M)*) of differential /?-forms of class C°° on
M is denoted by f_p(M) or by ^_P,_R(M). It is a module over <^(M), and is free
when T(M) is trivializable. We have ^"?(M) = <^j(M), and <f_p(M) may be
identified with the module of antisymmetric p-covariant tensor fields (16.18.4)
of class C°° over M.

Example

(16.20.2) Let /be a real-valued function of class C^r (r J> 1) on M, so that for
each point x e M the vector d_xfe T_X(M)* is a tangent covector at x (16.5.7).
Then the mapping xt-*d_xfis a differential form of class C¹""¹ on M (with the



	20 DIFFERENTIAL FORMS 137 where TT is the canonical mapping onto the orbit space (12.10.6) and j is a homeo- morphism onto a closed subset V_n of C^N. The canonical image of V_n — {0} in the pro- jective space P_N~i(C) is analytically isomorphic to P_n_i(C). (b) The complex manifold obtained by blowing up the point 0 in C^N (Section 16.11, Problem 3) is analytically isomorphic to the canonical bundle L_N__lfC(l). If

	is the canonical projection of this blowing-up (it is a local isomorphism everywhere outside the fiber q~^l(Q)), show that q~(V_n)* is analytically isomorphic to the bundle L_n_i,_c(2) over P_w_i(C) (Section 16.16, Problem 1).

	20. DIFFERENTIAL FORMS (16.20.1) Let M be a differential manifold. The dual T(M)* of the tangent bundle T(M) is called the cotangent bundle of M. If F is the transition diffeo- morphism between two charts on M, then, as we have seen, the transition diifeomorphism between the associated fibered charts of T(M) is

	(16.15.4.5). Hence the transition diffeomorphism between the associated fibered charts of T(M)* is (x, h)h+(F(), 'DF(jc)-¹ • h). We shall write TJ(M) in place of TJ(T(M)) when there is no risk of con- fusion; in particular, therefore, TJ(M) = T(M) and T?(M)=T(M). A section of TJ(M) over a subset A of M is called a tensor field* (or, by abuse of language, a tensor) of type (p, q) over A. The set F(M, T£(M)) of tensor fields of class C°° over M is denoted by «^~J(M) or ^_)R(M); it is a module over the ring rf(M) = <$(M ; R) (also denoted by <^_R(M)j of real-valued C°° -functions on M, and is &free module when T(M) is trivializable (16.15.8). For each p ^ 1, a section over A of the bundle /\T(M)* of tangent p- covectors is called a differential p-form on A (or simply a differential form when p = 1). The set F(M, y\T(M)) of differential /?-forms of class* C°° on M is denoted by f_p(M) or by ^_P,_R(M). It is a module over <^(M), and is free when T(M) is trivializable. We have ^"?(M) = <^j(M), and <f_p(M) may be identified with the module of antisymmetric p-covariant tensor fields (16.18.4) of class C°° over M. Example (16.20.2) Let /be a real-valued function of class C^r (r J> 1) on M, so that for each point x e M the vector d_xfe T_X(M)* is a tangent covector at x (16.5.7). Then the mapping xt-d_xfis* a differential form of class C¹""¹ on M (with the