TOTAL FOLIATIONS ON 3-MANIFOLDS AND THE GEOMETRY OF SPACES OF COFRAMEFIELDS (TOPOLOGY, DIFFERENTIAL)

If (alpha) is a 1-form on a 3-manifold M, the theorem of Frobenius implies that (alpha) is normal to a codimension-1 foliation of M iff (alpha)(WEDGE)d(alpha) = 0. If (alpha), (beta) and (gamma) are 1-forms on M satisfying (alpha)(WEDGE)(beta)(WEDGE)(gamma) = (omega) (NOT=) 0, then iff (alpha)(WEDGE)da = (beta)(WEDGE)d(beta) = (gamma)(WEDGE)d(gamma) = 0, (alpha), (beta) and (gamma) are normal to a total foliation, i.e. a triple of codimension-1 foliations everywhere mutually transversal.;I define G((alpha),(beta),(gamma)) =(INT) ((alpha)d(alpha))('2) + ((beta)d(beta))('2) + ((gamma)d(gamma))('2) (.)((alpha)(WEDGE)(beta)(WEDGE)(gamma))('-1) on the space X of all coframefields ((alpha),(beta),(gamma)) on M satisfying (alpha)(WEDGE)(beta)(WEDGE)(gamma) (NOT=) 0 and the normalization condition (INT)(alpha)(WEDGE)(beta)(WEDGE)(gamma) = 1. Constructions will be carried out either on this space X or on the subspace Z (L-HOOK) X defined by the additional normalization condition (VBAR)(alpha)d(alpha)(VBAR) = (VBAR)(beta)d(beta)(VBAR) = (VBAR)(gamma)d(gamma)(VBAR) = K(alpha)(WEDGE)(beta)(WEDGE)(gamma) for some constant K.;Chapter 3 I calculate the gradients of G on X and Z under appro- priately defined intrinsic Hilbert metrics. (DEL)(,X)G restricted to G gives the particularly simple form (DEL)(,X)G((alpha),(beta),(gamma)) = 4K(xi)('(alpha))d(alpha)* - 4K('2)(alpha), 4K(xi)('(beta))d(beta)* - 4K('2)(beta), 4K(xi)('(gamma))d(gamma)* - 4K('2)(gamma)), where (xi)('(alpha))(alpha)d(alpha) = (xi)('(beta))(beta)d(beta) = (xi)('(gamma))(gamma)d(gamma) =.;K(alpha)(WEDGE)(beta)(WEDGE)(gamma), (+OR-)(xi)('(alpha)) = (+OR-)(xi)('(beta)) = (+OR-)(xi)('(gamma)) = 1, and * is defined according to the unique metric in which ((alpha),(beta),(gamma)) is an orthonormal coframe. The critical points of G on X are shown to lie on Z; they are therefore the solutions of Kd(alpha)* = (+OR-)K('2)(alpha), Kd(beta)* = (+OR-)K('2)(beta), Kd(gamma)* = (+OR-)K('2)(gamma). This holds iff either (i) K = 0, hence G((alpha),(beta),(gamma)) = 0 and ((alpha),(beta),(gamma)) are normal to a total foliation; or (ii) d(alpha) = (+OR-)K(beta)(gamma), d(beta) = (+OR-)K(gamma)(alpha) = (+OR-)K(alpha)(beta), in which case the manifold M is shown to admit a transitive action of the universal covering group of either SO(3) or SO(2,1).;Detlef Hardorp has shown, by a complicated surgery construc- tion, that every closed oriented 3-manifold admits a total foliation. I approach the same problem by a variational method, using a modification of the method of steepest descent to construct critical points of a functional measuring the extent to which the conditions (alpha)d(alpha) = (beta)d(beta) = (gamma)d(gamma) = 0 fail to be satisfied.;It follows that unless M is a quotient manifold of Spin(3) or SO(2,1),(' )finding a critical point for G on Z (or X) is sufficient to construct a total foliation on M. In Chapter 4 I construct a family of curves (psi) in a Hilbert completion of Z satisfying the conditions (d/dt)G((psi)(t)) = -(VBAR)(d/dt) (psi)(t)(VBAR)('2) and (VBAR)(d/dt) (psi)(t)(VBAR) > CG((psi)(t))(' 1/2) for a given constant C, and show that under appropriate conditions they converge to critical points for G in the Hilbert completion of Z. These curves may be seen as approximate solutions to the differential equation (d/dt) (psi)(t) = -(DEL)G((psi)(t)), and so the method for finding critical points may be seen as a modification of the method of steepest descent. (Abstract shortened with permission of author.).