We prove that any special para-Kähler manifold is intrinsically an improper affine hypersphere. As a corollary, para-holomorphic function F of n para-complex variables satisfying non-degeneracy condition defines hypersphere, which the graph real f 2n variables. give explicit formula for in terms F. Necessary and sufficient conditions hypersphere to admit structure are given. Finally, it shown conical manifolds foliated by proper hyperspheres constant mean curvature.

We present a new general construction of examples mean curvature solitons on manifolds admitting nowhere-vanishing Killing vector field. Using Riemannian submersion techniques, we reduce the problem from PDE to an ODE. As application, obtain rotators in hyperbolic space.

In this work we study decompositions of para-complex and para-holomorphic vector-bundles endowed with a connection ∇ over manifold. First obtain results on the connections induced subbundles, their second fundamental forms curvature tensors. particular analyze decompositions. Then introduce notion affine immersions apply above to existence uniqueness theorems for immersions. This is generalization obtained by Abe Kurosu [AK] geometry. Further prove that any vanishing (0, 2)-curvature,...

In this paper we give a spinorial representation of submanifolds any dimension and codimension into Riemannian space forms in terms the existence so called generalized Killing spinors.We then discuss several applications, among them new concise proof fundamental theorem submanifold theory.We also recover results T. Friedrich, B. Morel authors 2 3.

Abstract We study new examples of translating solitons the mean curvature flow, especially in Minkowski space. consider for this purpose manifolds admitting submersions and cohomegeneity one actions by isometries on suitable open subsets. This general setting also covers classical Euclidean examples. As an application, we completely classify time-like, invariant rotations boosts

Abstract We examine which of the compact connected Lie groups that act transitively on spheres different dimensions leave unique spin structure sphere invariant. study notion invariance a and prove this classification in two ways; through examining differential actions representation theory.

We prove a Bonnet theorem for isometric immersions of submanifolds into the products an arbitrary number simply connected real space forms. Then, we existence associated families minimal surfaces in such products. Finally, case $\mathbb{S}^2\times\mathbb{S}^2$, give complex version main terms two canonical structures $\mathbb{S}^2\times\mathbb{S}^2$.

Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give characterization of three-dimensional strongly pseudoconvex pseudo-hermitian manifolds isometrically immersed in Euclidean space ℝn terms an integral representation Weierstraß type. Restricting to case immersions ℝ4, study harmonicity conditions for such complete classification CR-pluriharmonic immersions.

We recall the notion of (vertical) translating solitons in a product semi-Riemannian manifold $(M,g)$ and real line. Mainly, we restrict our attention to those which are graph smooth function. When dealing with submersions, show criteria lift (or project) from base total space viceversa). In particular, manifolds foliated by codimension 1 orbits Lie group action give rise such solitons, up solving first-order ordinary differential equation. This gives us explicit under function is soliton,...

We classify Riemannian surfaces admitting associated families in three dimensional homogeneous spaces with four-dimensional isometry groups and a wide family of (semi-Riemannian) warped products, an extra natural condition (namely, rotating structure vector field). prove that, provided the surface is not totally umbilical, such exist both cases if, only ambient manifold product minimal. In particular, there exists no field Heisenberg group.

Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give characterization of three-dimensional strongly pseudoconvex pseudo-hermitian CR-manifolds isometrically immersed in Euclidean space $\mathbb{R}^n$ terms an integral representation Weierstrass type. Restricting to case immersions $\mathbb{R}^4$, study harmonicity conditions for such complete classification CR-pluriharmonic immersions.

Spinorial methods have proven to be a powerful tool study geometric properties of spin manifolds. Our aim is continue the spinorial manifolds that are not necessarily spin. We introduce and notion $G$-invariance spin$^r$ structures on manifold $M$ equipped with an action Lie group $G$. For case when homogeneous $G$-space, we prove classification result these invariant in terms isotropy representation. As example, for all realisations spheres.

Abstract We present a method giving spinorial characterization of an immersion into product spaces constant curvature. As first application we obtain proof using spinors the fundamental theorem theory for such target spaces. also study special cases: recover previously known results concerning immersions in $$\mathbb {S}^2\times \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup>...