Abstract:
Conformal geometry is a weakening of Riemannian geometry where one works with a
smooth manifold equipped with an equivalence class of Riemannian metrics, where two
metrics are equivalent if and only if they define the same angles between curves. Early
interest in conformal equivalence included the question of biholomorphic equivalence of
domains in the complex plane. Interest has also been driven by physics and general relativity, since light in spacetime follows null geodesics, and these only depend on the conformal structure. Recently there has been considerable progress in the study of codimension
one submanifolds in conformal manifolds. At the other extreme, there has also been an
increased understanding of the distinguished curves in conformal manifolds. In this thesis,
we develop a complete basic tractor theory of conformal submanifolds of any codimension
and use this to define a notion of distinguished conformal submanifolds. These distinguished submanifolds coincide with conformal circles and totally umbilic hypersurfaces in
the extremal cases. We emphasize three conformal tractor objects which we show encode
equivalent submanifold data. Our notion of distinguished submanifolds admits characterizations in terms of all three invariants. Our definition immediately leads to a procedure
for proliferation of conserved quantities along these submanifolds. We also obtain a theorem which characterizes our distinguished conformal submanifolds in terms of an incidence
relation and a parallel condition. We use this to show that zero loci of certain solutions
to a conformally invariant equation are, if nonempty, distinguished submanifolds. These
results extend existing results for conformal circles.