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, .
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:
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 ] =
DVMn .
The Euler-Lagrange equations of (1.2) read:
= 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:
= 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
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
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).
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