arXiv research

Analyzes canonical reductive decomposition of extrinsic homogeneous submanifolds.

Classifies minimal submanifolds in complex hyperbolic spaces.

We consider parallel submanifolds $M$ of a Riemannian symmetric space $N$ and study the question whether $M$ is extrinsically homogeneous in $N$\,, i.e.\ whether there exists a subgroup of the isometry group of $N$ which acts transitively on $M$\,. First, given a "2-jet" $(W,b)$ at some point $p\in N$ (i.e. $W\subset T…

Classifies polar actions on 3D homogeneous spaces.

A subset S of a Riemannian manifold N is called extrinsically homogeneous if S is an orbit of a subgroup of the isometry group of N. Thorbergsson proved the remarkable result that every complete, connected, full, irreducible isoparametric submanifold of a finite dimensional Euclidean space of rank at least 3 is extrins…

Paper finds convexity in translating solitons for concave flows.

Study of Lagrangian submanifolds in pseudo-nearly Kähler SL(2,R)×SL(2,R).

Study non-degenerate anisocurved surfaces in homogeneous 3-manifolds.

Due to its length and several inaccuracies, this article is no longer suggested for reading. Moreover, in the meantime certain results presented herein could be improved by the author in a non-trivial fashion. Instead, the reader is referred to the following two articles: The extrinsic holonomy Lie algebra of a paralle…

We prove rigidity of oriented isometric immersions of complete surfaces in the homo- geneous 3-manifolds E(k; τ) (different from the space forms) having the same positive extrinsic curvature.

New systems of linear PDEs discovered in 3D contact manifolds.

Classification of hypersurfaces in homogeneous spaces with specific properties.

The homogeneous nearly Kähler structure on C⁴ is defined and analyzed for curvature and Lagrangian submanifolds.

Study on rolling Stiefel manifolds with specific metrics.

New homogeneous special Lagrangian submanifolds discovered in nearly Kähler CP3.

New classifications of totally real surfaces in nearly Kähler C⁴.

We prove a general extrinsic rigidity theorem for homogeneous varieties in $\mathbb{CP}^N$. The theorem is used to show that the adjoint variety of a complex simple Lie algebra $\mathfrak{g}$ (the unique minimal G orbit in $\mathbb{P}\mathfrak{g}$) is extrinsically rigid to third order. In contrast, we show that the ad…

Classifies totally geodesic submanifolds in Hopf-Berger spheres.

Consider a closed manifold $M$ immersed in $\R^m.$ Suppose that the trivial bundle $M\times\R^m=TM\otimes νM$ is equipped with an almost metric connection $\tilde{\nabla}$ which almost preserves the decomposition of $M\times\R^m$ into the tangent and the normal bundle. Assume moreover that the difference $Γ=\partial-\t…

Let $M$ be a simply connected homogeneous three-manifold with isometry group of dimension $4$, and let $Σ$ be any compact surface of genus zero immersed in $M$ whose mean, extrinsic and Gauss curvatures satisfy a smooth elliptic relation $Φ(H,K_e,K)=0$. In this paper we prove that $Σ$ is a sphere of revolution, provide…

The Gromoll-Meyer sphere is constructed and studied in a homogeneous space.

Constructing solutions to geometric flows with rotational symmetry.

We prove a descriptive theorem on the extrinsic geometry of an embedded minimal surface of injectivity radius zero in a homogeneously regular Riemannian three-manifold, in a certain small intrinsic neighborhood of a point of almost-minimal injectivity radius. This structure theorem includes a limit object which we call…

The study classifies surfaces in Berger spheres as Willmore and Hopf tori.

We give a unified method for the general equivalence problem of extrinsic geometry, on the basis of our formulation of a general extrinsic geometry as that of an osculating map $\varphi\colon (M,\mathfrak f) \to L/L^0 \subset \operatorname{Flag}(V,φ)$ from a filtered manifold $(M,\mathfrak f)$ to a homogeneous space $L…

The paper extends convexity results for translating solitons in higher dimensions.

Explicit formulas for extrinsic Paneitz operators and Q-curvatures for totally umbilic hypersurfaces.

The extrinsic Bonnet-Myers theorem is proven for positive Ricci curvature manifolds.

The study defines and characterizes extrinsic catenaries in hyperbolic space.

In this paper we prove two theorems. The first one is a structure result that describes the extrinsic geometry of an embedded surface with constant mean curvature (possibly zero) in a homogeneously regular Riemannian three-manifold, in any small neighborhood of a point of large almost-maximal curvature. We next apply t…

We describe extrinsic hyperspheres and totally geodesic hypersurfaces in manifolds with special holonomy. In particular we prove the nonexistence of extrinsic hyperspheres in quaternion-Kaehler manifolds. We develop a new approach to extrinsic hyperspheres based on the classification of special Killing forms.

We find a one-to-one correspondence between full extrinsic symmetric spaces in (possibly degenerate) inner product spaces and certain algebraic objects called (weak) extrinsic symmetric triples. In particular, this yields a description of arbitrary extrinsic symmetric spaces in pseudo-Euclidean spaces by corresponding …

Having developed a description of indefinite extrinsic symmetric spaces by corresponding infinitesimal objects in the preceding paper we now study the classification problem for these algebraic objects. In most cases the transvection group of an indefinite extrinsic symmetric space is not semisimple, which makes the cl…

Study on ratio of intrinsic to extrinsic metrics and its relation to surface area.

New operators and curvatures derived from embedded manifolds.

In this article we introduce a generalization of the Newton transformation to the case of a system of endomorphisms. We show that it can be used in the context of extrinsic geometry of foliations and distributions yielding new integral formulas containing generalized extrinsic curvatures.

This paper evaluates CFL algorithms for handling data heterogeneity in federated learning.

Novel coarse extrinsic curvature for Riemannian submanifolds.

Paper finds properties of special geometric shapes in higher dimensions.

We study infinitesimal semi-simple extrinsic symmetric spaces and give a classification in the symplectic case.

We study the topology of (properly) immersed complete minimal surfaces $P^2$ in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces, (see \cite{Pa}). We present an alternative and partially unified proof of…

We prove the energy identity and the no neck property for a sequence of smooth extrinsic polyharmonic maps with bounded total energy.

Extrinsic Geometric Flow (EGF) for a codimension-one foliation has been recently introduced by authors as deformations of Riemannian metrics subject to quantities expressed in terms of its second fundamental form. In the paper we introduce soliton solutions to EGF and study their geometry for totally umbilical foliatio…

Let U be a real form of a complex semisimple Lie group, and tau, sigma, a pair of commuting involutions on U. This data corresponds to a reflective submanifold of a symmetric space, U/K. We define an associated integrable system, and describe how to produce solutions from curved flats. The solutions are shown to corres…

In the paper we prove, that extrinsic curvature does not impose restrictions on the topology of a contact structure, except the obvious ones.

In this note, we prove lower and upper bounds for Dirac operators of submanifolds in certain ambient manifolds in terms of conformal and extrinsic quantities.

The diameter of a disc filling a loop in the universal covering of a Riemannian manifold may be measured extrinsically using the distance function on the ambient space or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram filling a word that represents the i…

We prove that complete submanifolds, on which the Omori-Yau weak maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower the codimension is, the richer such points are. The smaller the radius is, the la…