Hypersurfaces With a Common Geodesic Curve in 4D Euclidean space E4

Tangent Bundle of the Hypersurfaces in a Euclidean Space

Let $M$ be an orientable hypersurface in the Euclidean space $R^{2n}$ with induced metric $g$ and $TM$ be its tangent bundle. It is known that the tangent bundle $TM$ has induced metric $overline{g}$ as submanifold of the Euclidean space $R^{4n}$ which is not a natural metric in the sense that the submersion $pi :(TM,overline{g})rightarrow (M,g)$ is not the Riemannian submersion. In this paper...

Lorentzian Geodesic Flows between Hypersurfaces in Euclidean Spaces

There are several approaches to this question. One is from the perspective of a Riemannian metric on the group of diffeomorphisms of R. If the smooth hypersurfaces Mi bound compact regions Ωi , then the group of diffeomorphisms Diff(R) acts on such regions Ωi and their boundaries. Then, if φt, 1 ≤ t ≤ 1, is a geodesic in Diff(R) beginning at the identity, then φt(Ω) (or φt(Mi)) provides a path ...

Lk-BIHARMONIC HYPERSURFACES IN THE EUCLIDEAN SPACE

Chen conjecture states that every Euclidean biharmonic submanifold is minimal. In this paper we consider the Chen conjecture for Lk-operators. The new conjecture (Lk-conjecture) is formulated as follows: If Lkx = 0 then Hk+1 = 0 where x : M → R is an isometric immersion of a Riemannian manifold M into the Euclidean space R, Hk+1 is the (k+1)-th mean curvature of M , and Lk is the linearized ope...

Brownian Functionals on Hypersurfaces in Euclidean Space

Using the first exit time for Brownian motion from a smoothly bounded domain in Euclidean space, we define two natural functionals on the space of embedded, compact, oriented, unparametrized hypersurfaces in Euclidean space. We develop explicit formulas for the first variation of each of the functionals and characterize the critical points.

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form