For every N > 0 there exists a group of deficiency less than −N that arises as the fundamental group of a smooth homology 4–sphere and also as the fundamental group of the complement of a compact contractible submanifold of the 4–sphere. A group is the fundamental group of the complement of a contractible submanifold of the n–sphere, n > 4, if and only if it is the fundamental group of a homolo...

We investigate the estimation of a density f from n-sample on an Euclidean space RD, when data are supported by unknown submanifold M possibly dimension d<D, under reach condition. several nonparametric kernel methods, with data-driven bandwidths that incorporate some learning geometry via local estimator. When has Hölder smoothness ? and regularity ?, our estimator achieves rate n?????(2???+d)...

Abstract In this paper, we investigate any non-invariant submanifold of a locally decomposable golden Riemannian manifold in the case that rank set tangent vector fields induced structure on by ambient is less than or equal to codimension submanifold.

Abstract A decomposition theorem is established for a class of closed Riemannian submanifolds immersed in space form the constant sectional curvature. In particular, it shown that if M has nonnegative curvature and admits Codazzi tensor with “parallel mean curvature”, then locally isometric to direct product irreducible factors determined by spectrum tensor. This global when simply connected, g...

Let L be a special Lagrangian submanifold of R with boundary lying on the symplectic, codimension 2 submanifold W . It is shown how deformations of L which keep the boundary of L confined to W can be described by an elliptic boundary value problem, and two results about minimal Lagrangian submanifolds with boundary are derived using this fact. The first is that the space of minimal Lagrangian s...