Submanifolds with parallel normalized mean curvature vector in a unit sphere

Let M ⁿ be an n-dimensional closed submanifold of a sphere with parallel normalized mean curvature vector. Denote by S and H the squared norm of the second fundamental form and the mean curvature of M ⁿ , respectively. Assume that the fundamental group \({\pi_{1}(M^{n})}\) of M ⁿ is infinite and \({S\, \leqslant\, S(H)=n+\frac{n^{3}H^{2}}{2(n-1)}-\frac{n(n-2)H}{2(n-1)}\sqrt{n^{2}H^{2}+4(n-1)}}\), then S is constant, S = S(H), and M ⁿ is isometric to a Clifford torus \({S^{1}(\sqrt{1-r^{2}})\times S^{n-1}(r)}\) with \({r^{2}\leqslant \frac{n-1}{n}}\).     

Surfaces with zero mean curvature vector in neutral 4-manifolds

Space-like surfaces and time-like surfaces with zero mean curvature vector in oriented neutral 4-manifolds are isotropic and compatible with the orientations of the spaces if and only if their lifts to the space-like and the time-like twistor spaces respectively are horizontal. In neutral Kähler surfaces and paraKähler surfaces, complex curves and paracomplex curves respectively are such surfaces and characterized by one additional condition. In neutral 4-dimensional space forms, the holomorphic quartic differentials defined on such surfaces vanish. There exist time-like surfaces with zero mean curvature vector and zero holomorphic quartic differential which are not compatible with the orientations of the spaces and the conformal Gauss maps of time-like surfaces of Willmore type and their analogues give such surfaces.     

Complete submanifolds with parallel mean curvature vector in hyperbolic spaces

In this paper we proved a better estimate as well as generalized to higher codimensions of a theorem of Y.B. Shen on complete submanifolds with parallel mean curvature vector in a hyperbolic space.     

Biharmonic submanifolds with parallel mean curvature vector field in spheres

We present some results on the boundedness of the mean curvature of proper biharmonic submanifolds in spheres. A partial classification result for proper biharmonic submanifolds with parallel mean curvature vector field in spheres is obtained. Then, we completely classify the proper biharmonic submanifolds in spheres with parallel mean curvature vector field and parallel Weingarten operator associated to the mean curvature vector field.     

Classification of Lorentzian surfaces with parallel mean curvature vector in pseudo-Euclidean spaces

Spatial surfaces with parallel mean curvature vector in pseudo-Euclidean spaces of arbitrary dimension and index were classified by B.Y. Chen. In this work, we give a complete classification of Lorentzian surfaces with parallel mean curvature vector in pseudo-Euclidean spaces of arbitrary dimension and index. Consequently, the problem to classify all the surfaces with parallel mean curvature vector in pseudo-Euclidean spaces has been solved.     

Complete spacelike submanifolds with parallel mean curvature vector in a semi-Euclidean space

Our aim in this article is to study the geometry of n-dimensional complete spacelike submanifolds immersed in a semi-Euclidean space \({\mathbb{R}^{n+p}_{q}}\) of index q, with \({1\leq q\leq p}\). Under suitable constraints on the Ricci curvature and on the second fundamental form, we establish sufficient conditions to a complete maximal spacelike submanifold of \({\mathbb{R}^{n+p}_{q}}\) be totally geodesic. Furthermore, we obtain a nonexistence result concerning complete spacelike submanifolds with nonzero parallel mean curvature vector in \({\mathbb{R}^{n+p}_{p}}\) and, as a consequence, we get a rigidity result for complete constant mean curvature spacelike hypersurfaces immersed in the Lorentz–Minkowski space \({\mathbb{R}^{n+1}_{1}}\).     

A homogeneous submanifold with nonzero parallel mean curvature vector in a Euclidean sphere

We show that every sufficiently high dimensional Euclidean sphere admits an odd dimensional Riemannian submanifold M having the properties: (1) M is a homogeneous submanifold with nonzero parallel mean curvature vector in the ambient sphere; (2) M is a Berger sphere; (3) M is a Sasakian space form of constant ${\phi}$ -sectional curvature. Note that our manifold M is diffeomorphic but not isometric to a Euclidean sphere.     

Surfaces with constant mean curvature in Sol geometry

Among the eight geometries of Thurston, Sol₃ is the space with the smallest number of isometries, for example, there are no rotations. In this work, we classify all surfaces with constant mean curvature that either are invariant by a 1-parameter group of isometries or are the product of two planar curves.     

Submanifolds with prescribed mean curvature vector field

In a recent paper [2] K. Nomizu has shown that a natural analogue of an n-sphere in an arbitrary Riemannian manifold is an n-dimensional umbilical submanifold with non-zero parallel mean curvature vector, which he calls extrinsic sphere sometimes. This note is concerned with the question whether extrinsic spheres have a special topological or differentiable feature.     

A Note on surfaces with parallel mean curvature

We use a Simons type equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.     

Surfaces with harmonic inverse mean curvature in space forms

We define surfaces with harmonic inverse mean curvature in space forms and generalize a theorem due to Lawson by which surfaces of constant mean curvature in one space form isometrically correspond to those in another. We also obtain an immersion formula, which gives a deformation family for these surfaces.

     

Euclidean submanifolds with Jacobi mean curvature vector field

We study submanifolds in the Euclidean space whose mean curvature vector field is a Jacobi field. First, we characterize them and produce non-trivial (non-minimal) examples and then, we look for additional conditions which imply minimality.Research partially supported by a DGICYT grant No PB94-0705-C02-01 and by a grant of Gobierno Vasco PI95/95     

Complete classification of spatial surfaces with parallel mean curvature vector in arbitrary non-flat pseudo-Riemannian space forms

Submanifolds with parallel mean curvature vector play important roles in differential geometry, theory of harmonic maps as well as in physics. Spatial surfaces in 4D Lorentzian space forms with parallel mean curvature vector were classified by B. Y. Chen and J. Van der Veken in [9]. Recently, spatial surfaces with parallel mean curvature vector in arbitrary pseudo-Euclidean spaces are also classified in [7]. In this article, we classify spatial surfaces with parallel mean curvature vector in pseudo-Riemannian spheres and pseudo-hyperbolic spaces with arbitrary codimension and arbitrary index. Consequently, we achieve the complete classification of spatial surfaces with parallel mean curvature vector in all pseudo-Riemannian space forms. As an immediate by-product, we obtain the complete classifications of spatial surfaces with parallel mean curvature vector in arbitrary Lorentzian space forms.