Related papers: Submanifolds with constant Moebius curvature and f…

We define hypersurfaces $f\colon M^n\to \mathbb{Q}_{c_1}^{k} \times \mathbb{Q}_{c_2}^{n-k+1}$ in class $\mathcal{A}$ of a product of two space forms as those that have flat normal bundle when regarded as submanifolds of the underlying flat…

Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. They are Moebius invariant…

We classify smooth complex projective varieties $X \subset \proj^N$ of dimension $2s+1$ containing a linear subspace $\Lambda$ of dimension $s$ whose normal bundle $N_{\Lambda/X}$ is numerically effective.

Let $M$ and $N$ be Riemannian symmetric spaces and $f:M\to N$ be a parallel isometric immersion. We additionally assume that there exist simply connected, irreducible Riemannian symmetric spaces $M_i$ with $\dim(M_i)\geq 2$ for $i=1,...,r$…

In this paper, we consider a closed Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$, and a compact Lie group $G$ acting as isometries on $M$ with cohomogeneity at least $3$. Suppose the union of non-principal orbits…

We identify the smooth metrics $\mc{M}(M)$ on a manifold $M^n$ with the smooth isometric embeddings $f_g: (M,g) \rightarrow (\mb{S}^{\tn}, \tg)$ into a standard sphere of large dimension $\tn=\tn(n)$, and their Palais isotopic deformations,…

We present three ways to establish general stability inequalities for various classes of 2-immersions in Euclidean spaces of higher codimension

Submanifolds of Frobenius manifolds are studied. In particular, so-called natural submanifolds are defined and, for semi-simple Frobenius manifolds, classified. These carry the structure of a Frobenius algebra on each tangent space, but…

We give a necessary and sufficient condition for a 2-dimensional Riemannian manifold to be locally isometrically immersed into a 3-dimensional homogeneous manifold with a 4-dimensional isometry group. The condition is expressed in terms of…

We introduce a class of minimal submanfolds $M^n$, $n\geq 3$, in spheres $\mathbb{S}^{n+2}$ that are ruled by totally geodesic spheres of dimension $n-2$. If simply-connected, such a submanifold admits a one-parameter associated family of…

In this paper, we investigate complete curvature-adapted submanifolds with maximal flat section and trivial normal holonomy group in symmetric spaces of compact type or non-compact type under certain condition, and derive the constancy of…

A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean…

We classify compact oriented $5$-manifolds with free fundamental group and $\pi_{2}$ a torsion free abelian group in terms of the second homotopy group considered as $\pi_1$-module, the cup product on the second cohomology of the universal…

We consider the orthonormal frame bundle F(M) of a Riemannian manifold M. A construction of Sasaki defines a canonical Riemannian metric on F(M). We prove that for two closed Riemannian n-manifolds M and N, the frame bundles F(M) and F(N)…

Let M be a closed simply connected n-manifold of positive sectional curvature. We determine its homeomorphism or homotopic type if M also admits an isometric elementary p-group action of large rank. Our main results are: There exists a…

We give a diffeomorphism classification of pinched negatively curved manifolds with amenable fundamental groups, namely, they are precisely the M\"obius band, and the products of a line with the total spaces of flat vector bundles over…

In this article, we study constant mean curvature isometric immersions into $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ and we classify these isometric immersions when the surface has constant intrinsic curvature.…

We classify hypersurfaces of rank two of Euclidean space $\R^{n+1}$ that admit genuine isometric deformations in $\R^{n+2}$. That an isometric immersion $\hat f\colon\,M^n\to\R^{n+2}$ is a genuine isometric deformation of a hypersurface…

We define submersions f between manifolds M and N modelled on locally convex spaces. If the range N is finite-dimensional or a Banach manifold, then these coincide with the naive notion of a submersion. We study pre-images of submanifolds…

The complete local classification and geometric description of n-dimensional submanifolds F with recurrent nonparallel second fundamental form in the spaces of constant curvature M(c) are obtained in this article.