Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM have well-known physical interpretations and have been studied by physicists and geometers using some Riemannian and pseudo-Riemannian lift metrics on TM. Here we consider the Riemannian or pseudo-Riemannian lift metric G on TM which is in some senses more general than other lift metrics previously defined on TM, and seems to complete these works. Next we study the lift conformal vector fields on (TM, G).
In this article we study isometric immersions of nearly Kähler manifolds into a space form (specially Euclidean space) and show that every nearly Kähler submanifold of a space form has an umbilic foliation whose leafs are 6-dimensional nearly Kähler manifolds. Moreover using this foliation we show that there is no non-homogeneous 6-dimensional nearly Kähler submanifold of a space form. We prove some results towards a classification of nearly Kähler hypersurfaces in standard space forms.
In machine learning, a data set is often viewed as a point set distributed on a manifold. Using Euclidean norms to measure the proximity of this data set reduces the efficiency of learning methods. Also, many algorithms like Laplacian Eigenmaps or spectral clustering that require to measure similarity assume the k-Nearest Neighbors of any point are quite equal to the local neighborhood of the point on the manifold using Euclidean norms. In this paper, we propose a new method that intelligently transforms data on an unknown manifold to an n-sphere by the conformal stereographic projection, which preserves the angles and similarities of data in the original manifold. Therefore similarities represent actual similarities of the data in the original space. Experimental results on various problems, including clustering and manifold learning, show the effectiveness of our method.
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