Week 6
1. The Cotangent bundle of an m-manifold M.
A. Definition via transition functions. Consider M together with
a representative atlas {h_a: U_a -->
R^n} of coordinate charts. We have
seen that setting g_ab(x) = (Dx^a_i
/Dx^b_j)(x) defines a Gl(m,R)-valued
cocycle, and therefore an m-dimensional vector bundle which we have
identified with the tangent bundle.
(Remember that x^a_i
= x_i o h_a =
the i-th component of the h_a coordinate,
and that Df/Dx^b_j(x) =
D/Dx_j
(f o h_b^(-1))(h_b(x)).)
Defining g*_ab = g_ba^T (transpose)
gives a new cocycle, as can easily be checked. The corresponding
vector bundle is the cotangent bundle.
B. Definition via covectors. A covector at a point x in M is a linear map omega : TM_x --> R. The set of such linear maps is the vector space T*M_x, the co-tangent space at x. For example, if f is a smooth function defined near x, then f defines the covector df_x(v) = v(f). In particular, if x is in the coordinate chart h_a: U_a --> R^m, the m coordinate functions x^a_i = x_i o h_a define covectors dx^a_1,..., dx^a_m. These covectors form a basis for the cotangent space: if omega(D/Dx^a_i) = A^i, then omega = sum_i A^i dx^a_i. This allows the identification of T*U_a with U_a x R^m. In particular df = sum_i Df/Dx^a_i dx^a_i. Applying this last formula to the coordinates x^b_1 ,..., x^b_m of another chart h_b: U_b --> R^m defined near x gives dx^b_j = sum_i Dx^b_j/Dx^a_i dx^a_i. So if omega = sum_j B^j dx^b_j then omega = sum_j B^j sum_i Dx^b_j/Dx^a_i dx^a_i; comparing coefficients of dx^a_i yields A^i = sum_j B^j Dx^b_j/Dx^a_i, so the transition function relating the local trivializations over U_a and U_b is g*_ab = g_ba^T.
2. Vectorfields and 1-forms
Definition If p: E --> B is a (vector) bundle, a smooth section in E is a smooth map s: B --> E such that p o s = id_B. A section ``picks out a point in each fiber'' in a smooth way.
Definition A smooth section in TM --> M is a smooth vectorfield; a smooth section in T*M --> M is a smooth covectorfield, or more usually a smooth 1-form.
The bundles TM and T*M are topologically equivalent (see homework) but not in any natural way. There is an important law of composition between smooth vectorfields, called the Lie bracket, defined by [X,Y]f = X(Yf)-Y(Xf). Class exercise: show that [X,Y] is indeed a smooth vectorfield if X and Y are. But this law is non-associative, and does not permit useful computations with larger sets of vectorfields. We will now study an operation on forms; it is not a law of composition between covectorfields because the ``product'' of two 1-forms is a new kind of object called a 2-form.
First some multilinear algebra. Given a real vector space V, a linear p-form on V is a multilinear, alternating map K: V x ... x V --> R (p factors of V). Special cases: a linear 1-form is a linear map V --> R; if V has dimension n, then X_1 ,..., X_n --> det[X_1 ... X_n] (the determinant of the matrix whose ith row is X_i) is well known to be n-linear and alternating; a standard fact from linear algebra says that any n-linear alternating function V x ... x V --> R (n factors) is some constant multiple of the determinant; so the vector space of linear n-forms on V is 1-dimensional. Clearly the vector space of linear 1-forms is n-dimensional.
{Terminological note: I am using ``linear p-form'' here to distinguish this object from a smooth p-form on a manifold M, which will be a section in the bundle T*^pM --> M with fibre over x in M the vector space of linear p-forms on TM_x. Usually both are called simply ``p-form.''}
Proposition. If V has dimension n, the dimension of the vector space of linear p-forms on V is the binomial coefficient C(n,p).
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