Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high dimensional spaces. We consider the case where data is drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn” the homology of the submanifold with high confidence. We discuss an algorithm to do this and provide lear...

In the study of submanifolds in a homogeneous space X = G/P of a Lie group G, the method of moving frames is both a unifying concept and an effective tool, which is the version of the method of equivalence applied to submanifold geometry. Let φ be the left invariant, Lie algebra g-valued Maurer-Cartan form of G. The local equivalence problem for a submanifold f : M →֒ X is solved on a canonical ...

On an even-dimensional real submanifold of a Hermitian manifold, making use of the fundamental 2-form of the ambient manifold, we define a function. In this paper, we investigate the function in detail in some special submanifold.

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Given a sample from a probability measure with support on a submanifold in Euclidean space one can construct a neighborhood graph which can be seen as an approximation of the submanifold. The graph Laplacian of such a graph is used in several machine learning methods like semi-supervised learning, dimensionality reduction and clustering. In this paper we determine the pointwise limit of three d...

* Correspondence: meraj79@gmail. com Department of Mathematics, University of Tabuk, Tabuk, Kingdom of Saudi Arabia Full list of author information is available at the end of the article Abstract In this article, we have obtained necessary and sufficient conditions in terms of canonical structure F on a semi-invariant submanifold of an almost contact manifold under which the submanifold reduced...

We describe the asymptotic behavior of minimal area submanifolds in product spacetimes of an asymptotically hyperbolic space times a compact internal manifold. In particular, we find that unlike the case of a minimal area submanifold just in an asymptotically hyperbolic space, the internal part of the boundary submanifold is constrained to be itself a minimal area submanifold. For applications ...

We give a construction to obtain canonically an “isotropic average” of given C-close isotropic submanifolds of a symplectic manifold. To do so we use an improvement of Weinstein’s submanifold averaging theorem (obtained in collaboration with H. Karcher) and apply “Moser’s trick”. We also present an application to Hamiltonian group actions.