Related papers: Minimal two-spheres with constant curvature in the…

In this note we prove that a constant mean curvature surface is proper-biharmonic in the unit Euclidean sphere $\mathbb{S}^4$ if and only if it is minimal in a hypersphere $\mathbb{S}^3(\frac{1}{\sqrt{2}})$.

We study curvature-adapted submanifolds of general symmetric spaces. We generalize Cartan's theorem for isoparametric hypersurfaces of spheres and Wang's classification of isoparametric Hopf hypersurfaces in complex projective spaces to any…

Working in univalent foundations, we investigate the symmetries of spheres, i.e., the types of the form $\mathbb{S}^n = \mathbb{S}^n$. The case of the circle has a slick answer: the symmetries of the circle form two copies of the circle.…

In this note, we classify biharmonic submanifolds in a sphere defined by bi-eigenmaps ($\Delta^2 \phi=\lambda \phi$) or buckling eigenmaps ($\Delta^2 \phi=-\mu \Delta \phi$). We then classify biharmonic bi-eigenmaps and buckling eigenmaps…

We give a complete classification of the reductive symmetric pairs (G,H) for which the homogeneous space $(G \times H)/diag(H)$ is real spherical in the sense that a minimal parabolic subgroup has an open orbit. Combining with a criterion…

The $n$-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the $(n+1)$-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2-…

We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification…

Given a flat metric one may generate a local Hamiltonian structure via the fundamental result of Dubrovin and Novikov. More generally, a flat pencil of metrics will generate a local bi-Hamiltonian structure, and with additional…

Timelike minimal surfaces in the three-dimensional Lorentzian Heisenberg group are shown to be constructed from Lorentzian harmonic maps into the de-Sitter two-sphere, and they naturally admit singular points. In particular, we provide…

We give a geometric characterization of certain hypersurfaces of cohomogeneity one in the complex projective and hyperbolic planes. We also obtain some partial classifications of austere hypersurfaces and of Levi-flat hypersurfaces with…

We prove that a four-dimensional Lorentzian manifold that is curvature homogeneous of order 3, or $\CH_3$ for short, is necessarily locally homogeneous. We also exhibit and classify four-dimensional Lorentzian, $\CH_2$ manifolds that are…

We present an explicit characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of compact two-point homogeneous spaces, in the cases in which at least one of the spaces is a sphere of…

We introduce the concept of a standard form for two embedded maximal sphere systems in the doubled handlebody, and we prove an existence and uniqueness result. In particular, we show that pairs of maximal sphere systems in the doubled…

Let $G$ be a compact Lie group acting isometrically on a compact Riemannian manifold $M$ with nonempty fixed point set $M^G$. We say that $M$ is fixed-point homogeneous if $G$ acts transitively on a normal sphere to some component of $M^G$.…

The geometry of conformal minimal two-spheres immersed in G(2,6;R) is studied in this paper by harmonic maps. Then in most cases, we determine the linearly full reducible conformal minimal immersions from S^2 to G(2,8;R) identified with the…

The complement of a complex hyperplane arrangement is known to be homotopic to a minimal CW complex. There are several approaches to the minimality. In this paper, we restrict our attention to real two dimensional cases, and introduce the…

Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then…

For all spherical homogeneous spaces G/H, where G is a simply connected semisimple algebraic group and H a connected solvable subgroup of G, we compute the spectra of the representations of G on spaces of regular sections of homogeneous…

We give a complete classification of isomorphism classes of finitely generated projective modules, or equivalently, unitary equivalence classes of projections, over the C*-algebra $C\left( \mathbb{S}_{q}^{2n+1}\right) $ of the quantum…

We call a quaternionic Kaehler manifold with non-zero scalar curvature, whose quaternionic structure is trivialized by a hypercomplex structure, a hyper-Hermitian quaternionic Kaehler manifold. We prove that every locally symmetric…