10

C. TAUBE S

does no t com e equippe d wit h a natura l splittin g excep t alon g th e zer o

section, s 0, sinc e ds

0

: TX — • TV\

So

give s a canonical splitting . (Th e

zero sectio n o f a vecto r bundl e V assign s t o eac h x E X th e origi n i n

the vecto r spac e V\ x.)

A connectio n A o n V i s , among othe r things , a splittin g o f (1.21),

that is , it i s a linear map

(1.22) A : TV — » n*V

which obey s A o i = identity . Furthermore , A i s required t o be homo -

geneous o f degre e on e with respec t t o th e natura l actio n o f K = E o r

C o n V. Tha t is , K act s o n V b y multiplicatio n a s a subalgebr a o f

End(V). Denot e this actio n b y mv :KxV — • V. Likewise , K acts on

ir*V an d m^v wil l be used t o denote th e latter action . Fo r A G K , th e

map ray (A, •) can be used t o pull bac k A a s a 1-form o n V wit h value s

in

TT*V.

(Not e tha t ray(A, -)*7r* y i s naturally isomorphi c t o

TT*V

sinc e

7r o ?7iv(A, •) = 7r. ) Alternately , m

1x

*v{\, • ) can be used t o multipl y th e

value o f A o n a give n vector . Thes e tw o actions o f an elemen t A £ K

should agree . Tha t is ,

(1.23) m y(A, -)*A = ra*.v(A, A)

Covariant derivative s an d connection s ar e related i n th e sens e tha t

one i s completely equivalen t t o th e other. Indeed , i f s : V — • X i s a

section, the n s*A defines a 1-form on X wit h value s in V. And , it is an

exercise t o chec k tha t th e assignmen t t o a sectio n 5 of the 1-form s* A

defines a covariant derivative , VA , o n V.

Likewise, if a covariant derivativ e V has been defined fo r V, the n one

can defin e a connection o n V a s follows: Becaus e Aoi = identity , i t is

enough t o evaluate A o n vectors o f the form ds • v wher e s is a sectio n

of V an d where v is a vector i n TX. (Here , ds is the differential o f s as

a ma p from X t o V.) Wit h thi s understood , se t

(1.24) A(ds\

x

• v) = (six), (Vs)(v)) G 7r*y|s(x),

where x = n(v).

2. T H E ANTI-SEL F DUA L EQUATION S

This secon d lectur e ha s two purposes: I t give s a broader an d deepe r

introduction t o the anti-self dua l equations for a metric compatible con -

nection o n a comple x vecto r bundle . And , i t give s a n outlin e o f th e

celebrated application s of these equation s b y Donaldson and , now, oth-

ers.

Here, unti l furthe r notice , X wil l denot e a n oriented , Riemannia n

4-manifold. Tha t is , th e metri c wil l b e fixed. And , V wil l usuall y

http://dx.doi.org/10.1090/cbms/089/02