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Kang-Hai Tan

Publications and source records attributed to Kang-Hai Tan.

4 recordsLinked to original sources
arXiv2007

On some sub-Riemannian objects in hypersurfaces of sub-Riemannian manifolds

We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isolated characteristic points, then given two points, there exists at least one piecewise smooth horizontal curve in this hypersurface connecting them. In any sub-Riemannian manifold, we obtain the sub-Riemannian version of the fundamental theorem of Riemannian geometry. We use this nonholonomic connection to study horizontal mean curvature of hypersurfaces.

math.DG↗
arXiv2007

Convex Functions on Sub-Riemannian Manifolds. I

We find a different approach to define convex functions in the sub-Riemannian setting. A function on a sub-Riemannian manifold is nonholonomically geodesic convex if its restriction to any nonholonomic (straightest) geodesic is convex. In the case of Carnot groups, this definition coincides with that by Danniell-Garofalo-Nieuhn (equivalent to that by Lu-Manfredi-Stroffolini). Nonholonomic geodesics are defined using the horizontal connection. A new distance corresponding to the horizontal connection has been introduced and near regular points proven to be equivalent to the Carnot-Carathèodory distance. Some basic properties of convex functions are studied. In particular we prove that any nonholonomically geodesic convex function locally bounded from above is locally Lipschitzian with respect to the Carnot-Carathèodory distance.

math.DG↗
arXiv2007

On sublaplacians of sub-Riemannian manifolds

In this note we address a notion of sublaplacians of sub-Riemannian manifolds. In particular for fat sub-Riemannian manifolds we answered the sublaplacian question proposed by R. Montgomery.

math.DG↗