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Pawel Walczak

Publications and source records attributed to Pawel Walczak.

9 recordsLinked to original sources
arXiv2022

On isometric immersions of almost $k$-product manifolds

A Riemannian manifold endowed with $k\ge2$ complementary pairwise orthogonal distributions is called a Riemannian almost $k$-product manifold. In the article, for the first time, we study the following problem: find a relationship between intrinsic and extrinsic invariants of a Riemannian almost $k$-product manifold isometrically immersed in another Riemannian manifold. For such immersions, we establish an optimal inequality that includes the mixed scalar curvature and the square of the mean curvature. Although Riemannian curvature tensor belongs to intrinsic geometry, a special part called the mixed curvature is also related to the extrinsic geometry of a Riemannian almost $k$-product manifold. Our inequality also contains mixed scalar curvature type invariants related to B.-Y Chen's $δ$-invariants. Applications are given for isometric immersions of multiply twisted and warped products (we improve some known optimal inequalities by replacing the sectional curvature with our invariant) and to problems of non-immersion and non-existence of compact leaves of foliated submanifolds.

math.DG↗
arXiv2020

Deforming convex bodies in Minkowski geometry

We introduce and study deformation $T_{{\bf b},ϕ}$ of Minkowski norms in $\mathbb{R}^n$, determined by a set ${\bf b}=(β_1,\ldots,β_p)$ of linearly independent 1-forms and a smooth positive function $ϕ$ of $p$ variables. In particular, the $T_{{\bf b},ϕ}$-image of a Euclidean norm $α$ is a Minkowski norm, whose indicatrix is a rotation hypersurface with a $p$-dimensional axis passing through the origin. For $p=1$, our deformation generalizes construction of $(α,β)$-norm; the last ones form a rich class of "computable" Minkowski norms and play an important role in Finsler geometry. We use compositions of $T_{{\bf b},ϕ}$-deformations with ${\bf b}$'s of length $p$ to define an equivalence relation $\overset{p}\sim$ on the set of all Minkowski norms in $\mathbb{R}^n$. We apply M. Matsumoto result to characterize the cases when the Cartan torsions of a norm and its $T_{{\bf b},ϕ}$-image either coincide or differ by a $C$-reducible term.

math.DG↗
arXiv2018

A Godbillon-Vey type invariant for a 3-dimensional manifold with a plane field

We consider a 3-dimensional smooth manifold $M$ equipped with an arbitrary, \textit{a priori} non-integrable, distribution (plane field) ${\cal D}$ and a vector field $T$ transverse to ${\cal D}$. Using a 1-form $ω$ such that ${\cal D} = \ker\,ω$ and $ω(T)=1$ we construct a 3-form analogous to that defining the Godbillon-Vey class of a foliation, and show how does this form depend on $ω$ and~$T$. For a compatible Riemannian metric on $M$, we express this 3-form in terms of the curvature and torsion of normal curves and the non-symmetric second fundamental form of ${\cal D}$. We deduce Euler-Lagrange equations of associated functionals: for variable $({\cal D},T)$ on $M$, and for variable Riemannian or Randers metric on $(M,{\cal D})$. We show that for a geodesic field $T$ (e.g., for a contact structure) such $({\cal D},T)$ is critical, characterize critical pairs when ${\cal D}$ is integrable, and prove that these critical pairs are not extrema.

math.DG↗
arXiv2011

Extrinsic geometric flows on foliated manifolds, I

We study deformations of Riemannian metrics on a given manifold equipped with a codimension-one foliation subject to quantities expressed in terms of its second fundamental form. We prove the local existence and uniqueness theorem and estimate the existence time of solutions for some particular cases. The key step of the solution procedure is to find (from a system of quasilinear PDE's) the principal curvatures of the foliation. Examples for extrinsic Newton transformation flow, extrinsic Ricci flow, and applications to foliations on surfaces are given.

math.DG↗
arXiv2010

Extrinsic geometric flows on foliated manifolds, II

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 foliations, foliations on surfaces, and when the EGF is produced by the extrinsic Ricci tensor.

math.DG↗
arXiv2010

Extrinsic curvatures of distributions of arbitrary codimension

In this article, using the generalized Newton transformation, we define higher order mean curvatures of distributions of arbitrary codimension and we show that they agree with the ones from Brito and Naveira (Ann. Global Anal. Geom. 18, 371-383 (2000)). We also introduce higher order mean curvature vector fields and we compute their divergence for certain distributions and using this we obtain total extrinsic mean curvatures.

math.DG↗
arXiv2009

Dynamical behavior of Darboux curves

In 1872 G. Darboux defined a family of curves on surfaces of R^3 which are preserved by the action of the Mobius group and share many properties with geodesics. Here we characterize these curves under the view point of Lorentz geometry and prove some general properties and make them explicit them on simple surfaces, retrieving results of Pell (1900) and Santalo (1941).

math.DG↗
arXiv2009

Conformal fields and the stability of leaves with constant higher order mean curvature

In this paper, we study submanifolds with constant $r$th mean curvature $S_r$. We investigate, the stability of such submanifolds in the case when they are leaves of a codimension one foliation. We also generalize recent results by Barros - Sousa and Alías - Colares, concerning conformal fields, to an arbitrary manifold. Using this we show that normal component of a Killing field is a $r$th Jacobi field of a submanifold with $S_{r+1}$ constant. Finally, we study relations between $r$th Jacobi fields and vector fields preserving a foliation.

math.DG↗