2 1. INTRODUCTION

We may also endow the fibers g of V with an Ad(G) invariant metric ·, · which

respects the action of D. That is, one has the formula:

(1.1) d φ, ψ = Dφ, ψ + φ, Dψ .

The curvature associated to D is the g valued two-form F which arises from the

commutation of covariant derivatives and is defined via the formula:

DX DY φ − DY DX φ − D[X,Y

]

φ = [F (X, Y ),φ] .

We say that the connection D satisfies the Yang-Mills equations if its curvature is

a (formal) stationary point of the following Maxwell type functional:

(1.2) L[F ] = −

1

4

Mn

Fαβ,F

αβ

DVMn .

The Euler-Lagrange equations of (1.2) read:

(1.3)

DβFαβ

= 0 .

Furthermore, from the fact that F arises as the curvature of some connection D

the “Bianchi identity” is satisfied:

(1.4) D[αFβγ] = 0.

From now on we will refer to the system (1.3)–(1.4) as the first order Yang–Mills

equations (FYM).

Our aim is to study the Cauchy problem for the FYM system. To describe this

in a geometrically invariant way, we split the connection-curvature pair (F, D) in

the following way: Foliating M by the standard Cauchy hypersurfaces t = const.,

we decompose:

(F, D) = (F , D) ⊕ (E, D0) ,

where (F , D) denotes the portion of (F, D) which is tangent to the surfaces t =

const. (i.e. the induced connection), and (E, D0) denotes respectively the interior

product of F with the foliation generator T = ∂t, and the normal portion of D. In

inertial coordinates we have:

Ei = F0i , D0 = D∂t .

On the initial Cauchy hypersurface t = 0 we call a set

(F(0),D(0),E(0)) admissible

Cauchy data1 if it satisfies the following compatibility condition:

(1.5)

DiEi(0)

= 0 .

We define the Cauchy problem for the Yang-Mills equation to be the task of con-

struction a connection (F, D) which solves (1.3), and has Cauchy data equal to

(F(0),D(0),E(0)).

In order to understand what the appropriate conditions on the initial data

should be, it is necessary to consider the following two basic features of the system

1Note

that the set of initial data (F (0), D(0), E(0)) is overdetermined because the initial

curvature F (0) depends completely on the initial connection D(0). Also note while it is not

completely obvious at first that the set of initial data uniquely determines a solution (F, D) to

(1.3)–(1.4), this is the case. In particular, it is not necessary to specify the normal derivative

D0(0) initially as long as one knows the initial normal curvature E(0). This is a consequence

of the constraint equation (1.5) as will be demonstrated shortly (see equations (3.19) and (3.20)

below, and the discussion following).