Some Aspects Of Geometry And Analysis In Sub-Riemannian Manifolds

For H-type groups, explicit equations of shortest sub-Riemannian geodesies and a full characterization of the sub-Riemannian isometry groups are given in this dissertation. For a sub-Riemannian manifold (M, â–³ ,gc), the author proved that there exists a unique nonholonomic connection completely determined by the sub-Riemannian structure ( â–³ , gc). This is a sub-Riemannian version of the fundamental theorem in Rie-mannian geometry. The author used this intrinsic connection to give the notion of horizontal mean curvatures of hypersurfaces in sub-Riemannian manifolds.Generalized harmonic maps from Carnot-Caratheodory spaces (Rn,â–³, gc) to separable metric spaces were studied in this dissertation. The author proved a general existence theorem when the domain is smooth and noncharacteristic. When the target is of nonpositive curvature in the sense of Alexandrov, the noncharacteristic condition of the domain was removed and the uniqueness and local Holder continuity of energy minimizers were obtained. If moreover the domain is a smooth open set in a Carnot group, the energy minimizer is locally Lipschitz continuous. When the target is the Heisenberg group and the domain is in R2, partial regularity for energy minimizers were obtained.