Differential Geometry

In this chapter we compile several results from differential geometry which will be used throughout these notes. We advise the reader to use this chapter for reference purposes only. Most of the material is either part of or easily derived from a graduate course in differential geometry using, for example, F. Warner [27] or R. Hermann [9] as a text.

1. The De Rham Theorem

Let M be a paracompact, connected, C8 manifold. We denote by

Dk(M) (resp. Dk(M, C))

the differential forms (resp. complex valued) differential forms of degree k on M. We will use the notation Dk if the context M (or C) is clear.

Let d: Dk [right arrow] Dk+1 be the operation of exterior differentiation. The de Rham cohomology "groups" of M are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

k = 0, 1, 2, ..., D-1 = (0).

De Rham's theorem relates Hkd(M; K) with Hk(M; K) where Hk(M; K) is one of the usual cohomology theories on M (we will use the Cech cohomology theory). Since we will need an explicit form of the de Rham theorem in the next few chapters we review in detail the isomorphisms Hkd (M; K) with Hk(M; K), k = 0, 1, 2. (A complete account of the de Rham theorem can be found in F. Warner [27].)

We recall the definition of the Cech cohomology groups. Let G be an abelian group written additively. If

U = [Ua}a[member of]I

is an open covering of M, then a U-k-cochain on M with values in G is a rule, c, that assigns to each collection

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

of elements in U such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

an element [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let Ck(U; G) be the set of all U-k-cochains with values in G. If c, f [member of] Ck(U; G) then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

defines an abelian group structure on Ck(U; G).

Define d: Ck(U; G) [right arrow] Ck+1 (U; G) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(here the "roof" over an index means delete the index). It is easy to see that d2 = 0. Set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

h = 0, ... (C-1(U; G) = (0)).

Let U and V be open coverings of M. Then we use the notation:

U > V if U refines V ¦

If U >V, choose t: U [right arrow] V so that if X [member of] U then t X [subset] X. If c [member of] Ck (V; G) then define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then t* o d = d o t*. This implies that t* induces a map

t*: Hk(V; G) [right arrow] Hk(U; G).

t* is independent of the choice of t(c.f. Hirzburch [10]). Set tUV equal to the map

t*: Hk(V; G) [right arrow] Hk(U; G).

Then, if V< W< U,

tWV o tUW = tUV

by the above observations. We define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

That is, form the disjoint union of the Hk(U; G). If a [member of] Hk(U; G) and b [member of] Hk(V; G) then we say:

a ~ b

if there is

W > U and W > V

such that

tWUa = tWVb.

Then a ~ b is an equivalence relation and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the set of all equivalence classes. Hk(M; G) is an abelian group under the obvious operations.

A C8 map of [0, ] x M [right arrow] M is a continuous map that extends to a C8 map of (-e, 1 + e) x M [right arrow] M for some e > 0.

Definition 1.1. If U [subset] M is open, then we say that U is (C8) contractible if there is a C8 map

F: [0, 1] x U [right arrow] U

so that F(0, x) = x for all x [member of] U and F(1, x) = x0 for all x [member of] U. An open covering U = {Ua}a[member of]I is said to be contractible if, whenever

a1, ..., ak [member of] I

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

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is contractible.

Theorem 1.2.If U is an open covering of M then U has a contractible refinement.

This result is proved by putting a Riemannian structure on M and taking for each Ua [member of] U, p [member of] Ua, Np,U a convex neighborhood of p in U.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a contractible refinement. See Helgason [8] for the existence (and definition of convex neighborhoods).

Theorem 1.3.

a) If U is a contractible covering of M then Hk(U; G) is isomorphic with Hk(M; G).

b) If G is the additive group of R or C we denote Hk (M; G) by Hk(M; R) or Hk(M; C). Hk(M; G) is canonically isomorphic with Hkd(M; G). We denote the isomorphism by

d x R: Hkd(M; K) [right arrow] Hk(M; K).

Of course, this theorem is well known. a) is due to A. Weil [29] and b) is the de Rham theorem. We give a proof of b) (and a) for K = R or C) and for k = 0, 1 or 2 since we will be using explicit forms of the isomorphisms asserted to exist in b). We need a special case of Theorem 1.3.

Lemma 1.4 (Poincaré Lemma).If U [subset] M is an open contractible set and ? [member of] Dk, d? = 0, then there is ? [member of] Dk-1 so that ? = d?.

We will now give a sketch of a proof of this lemma. Let U be an open subset of a smooth manifold M and let F: [0, 1] x U [right arrow] U be C8. If 0 = t = 1 then we set Ft(u) = F(t, u) for u [member of] U. We will identify ? [member of] Dk ([0, 1] x U) (that is, ? is the restriction of an element of Dk ((-e, 1 + e) x U)) with an element of Dk(U) if it satisfies the two conditions:

1. for all v1, ..., vk [member of] T(t, p) ((-e, 1 + e) x M)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. If we take local coordinates x1, ..., xn in a neighborhood, Z, of p in U and local...