A Bridge between Complex Geometry and Riemannian Geometry

A Geometry Preserving Kernel over Riemannian Manifolds

Abstract- Kernel trick and projection to tangent spaces are two choices for linearizing the data points lying on Riemannian manifolds. These approaches are used to provide the prerequisites for applying standard machine learning methods on Riemannian manifolds. Classical kernels implicitly project data to high dimensional feature space without considering the intrinsic geometry of data points. ...

Riemannian geometry

Discrete Riemannian Geometry

Within a framework of noncommutative geometry, we develop an analogue of (pseudo) Riemannian geometry on nite and discrete sets. On a nite set, there is a counterpart of the continuum metric tensor with a simple geometric interpretation. The latter is based on a correspondence between rst order di erential calculi and digraphs (the vertices of the latter are given by the elements of the nite se...

Liouville and Carathéodory Coverings in Riemannian and Complex Geometry

A Riemannian manifold resp. a complex space X is called Liouville if it carries no nonconstant bounded harmonic resp. holomorphic functions. It is called Carathéodory, or Carathéodory hyperbolic, if bounded harmonic resp. holomorphic functions separate the points of X . The problems which we discuss in this paper arise from the following question: When a Galois covering X with Galois group G ov...

A Course in Riemannian Geometry