Quasi-Natural Symmetries and Connections for Bilinear Forms

Abstract

In this dissertation, we explore a different approach to the construction of connections that parallelize bilinear forms. The approach starts by generalizing the symmetries on the tangent bundle through the study of the maps that are bundle isomorphism between the two vector bundle structures on the T TM, that is the vector bundle structure associated with tangent bundle structure π ′ : T TM → TM and the induced structure π ∗ : T TM → TM determined by π : TM → M. The study of a set of involutive maps that are referred to as quasi-natural symmetries leads to a new method for constructing connections. Quasi-natural symmetries, although they are not truly natural, preserves a certain natural subbundle of T TM, namely V TM ∩T(TM)0. Connections arise within this structure because quasi-natural symmetries can be characterized in terms of their parities, where the symmetries of opposite parities commute and this property leads to the construction of connections. To understand connections that parallelize a bilinear from this perspective the category of symmetries is enlarged to include anti-symmetries, which are involutive maps of T TM that preserve the π ′ and π∗ structures up to a sign. It is shown that nondegenerate bilinear forms generated unique diffeomorphism of T TM that when the form is symmetric are a symmetries of odd parity and when the form is skew-symmetric are an anti-symmetries of odd parity. The symmetries generated by the bilinear forms can be used to construct parallelizing connections. The symmetric case is well understood, but the method can be used to completely characterize the connections that parallelize an almost symplectic form. It is shown that connections parallelizing an almost symplectic form can be parameterized by a real parameter and sections of the bundle of 3-tensors that can be completely described in terms of the natural decomposition of the 3-tensors by the canonical Young projectors. These methods lead to a new perspective on the difference between the connection theories in the symmetric and skew-symmetric cases. In the symmetric case because the symmetry associated with the bilinear form is of opposite parity to the natural symmetry parallelizing connections are uniquely determined up to torsion. In the skew case, the natural anti-symmetry is of the same parity as the anti-symmetry generated by the bilinear form and so an additional structure needs to be specified to obtain a parallelizing connection. This difference leads to the need for a background structure in specifying a connection that parallelizes an almost symplectic form as is seen in the literature.