arXiv subjects

Tomasz Zawadzki

Publications and source records attributed to Tomasz Zawadzki.

8 recordsLinked to original sources

arXiv2026

Non-canonical variations of Riemannian submersions with totally geodesic fibers

Using variations of Riemannian metric that preserve a given Riemannian submersion, keep its fibers totally geodesic and the metric restricted to the fibers fixed, but change the horizontal distribution, we examine changes of sectional curvatures in horizontal and vertical directions. We obtain conditions, in terms of a 1-form defining a variation, to locally make all sectional curvatures positive on the product of a manifold with positive curvature and a circle, while preserving the Riemannian submersion with geodesic fibers defined by the projection from the product. We examine conditions for obtaining weak contact metric structures from K-contact structures. We demonstrate existence of fat Riemannian submersions with totally geodesic fibers and vertizontal (i.e., spanned by a horizontal and a vertical vector) curvatures non-constant along a fiber. For a Riemannian submersion defined by an isometric group action, with totally geodesic fibers of dimension higher than one, we find variations that preserve the isometric action, while changing the horizontal distribution and its curvatures.

math.DG↗

arXiv2025

Variations of metric that preserve a Riemannian submersion and geometry of its fibers

On the domain of a Riemannian submersion, we consider variations (i.e., smooth one-parameter families) of Riemannian metrics, for which the submersion is Riemannian and which all keep the metric induced on its fibers fixed. We obtain a formula for the variation of the second fundamental form of the fibers with respect to such changes of metric. We find a choice of parameters defining the variations, that allows to easily formulate the necessary and sufficient conditions for preserving particular geometry of the fibers, i.e., keeping them totally geodesic, totally umbilical, or minimal. These conditions are related to the existence of Killing, conformal Killing and divergence-free vector fields on the fibers. We find conditions for metric to be a critical point of integrated squared norms of the mean curvature and the second fundamental form of the fibers, with respect to the considered variations, and prove that at all critical points of these functionals the second variation is non-negative. We also examine variations of sectional curvatures of planes defined by the horizontal lifts of vectors from the image of the submersion. In particular, we find that some variations preserving Riemannian submersions with totally geodesic fibers can make vertizontal curvatures not constant on the fibers.

math.DG↗

arXiv2024

Clairaut Conformal Submersions

The aim of this paper is to introduce Clairaut conformal submersions between Riemannian manifolds. First, we find necessary and sufficient conditions for conformal submersions to be Clairaut conformal submersions. In particular, we obtain Clairaut relation for geodesics on the total manifolds of conformal submersions, and prove that Clairaut conformal submersions have constant dilation along their fibers, which are totally umbilical, with mean curvature being gradient of a function. Further, we calculate the scalar and Ricci curvatures of the vertical distributions of the total manifolds. Moreover, we find a necessary and sufficient condition for Clairaut conformal submersions to be harmonic. For a Clairaut conformal submersion we find conformal changes of the metric on its domain or image, that give a Clairaut Riemannian submersion, a Clairaut conformal submersion with totally geodesic fibers, or a harmonic Clairaut submersion. Finally, we give two non-trivial examples of Clairaut conformal submersions to illustrate the theory and present a local model of every Clairaut conformal submersion with integrable horizontal distribution.

math.DG↗

arXiv2023

Variations of the mutual curvature of two orthogonal non-complementary distributions

On a smooth manifold with distributions ${\cal D}_1$ and ${\cal D}_2$ having trivial intersection, we consider the integral of their mutual curvature, as a functional of Riemannian metrics that make the distributions orthogonal. The mutual curvature is defined as the sum of sectional curvatures of planes spanned by all pairs of vectors from an orthonormal basis, such that one vector of the pair belongs to ${\cal D}_1$ and the second vector belongs to ${\cal D}_2$. As such, it interpolates between the sectional curvature of a plane field (if both distributions are one-dimensional), and the mixed scalar curvature of a Riemannian almost product structure (if both distributions together span the tangent bundle). We derive Euler-Lagrange equations for the functional, formulated in terms of extrinsic geometry of distributions, i.e., their second fundamental forms and integrability tensors. We give examples of critical metrics for distributions defined on domains of Riemannian submersions, twisted products and $f$-$K$-contact manifolds.

math.DG↗

arXiv2023

Variational problem on a metric-affine almost product manifold

We study a variational problem on a smooth manifold with a decomposition of the tangent bundle into $k>2$ subbundles (distributions), namely, we consider the integrated sum of their mixed scalar curvatures as a functional of adapted pseudo-Riemannian metric (keeping the pairwise orthogonality of the distributions) and contorsion tensor, defining a linear connection. This functional allows us to generalize the class of Einstein metrics in the following sense: if all of the distributions are one-dimensional, then it coincides with the geometrical part of the Einstein-Hilbert action restricted to adapted metrics. We prove that metrics in pairs metric-contorsion critical for our functional make all of the distributions totally umbilical. We obtain examples and obstructions to existence of those critical pairs in some special cases: twisted products with statistical connections; semi-symmetric connections and 3-Sasaki manifolds with metric-compatible connections.

math.DG↗

arXiv2020

The Einstein-Hilbert type action on metric-affine almost-product manifolds

We continue our study of the mixed Einstein-Hilbert action as a functional of a pseudo-Riemannian metric and a linear connection. Its geometrical part is the total mixed scalar curvature on a smooth manifold endowed with a distribution or a foliation. We develop variational formulas for quantities of extrinsic geometry of a distribution on a metric-affine space and use them to derive Euler-Lagrange equations (which in the case of space-time are analogous to those in Einstein-Cartan theory) and to characterize critical points of this action on vacuum space-time. Together with arbitrary variations of metric and connection, we consider also variations that partially preserve the metric, e.g., along the distribution, and also variations among distinguished classes of connections (e.g., statistical and metric compatible, and this is expressed in terms of restrictions on contorsion tensor). One of Euler-Lagrange equations of the mixed Einstein-Hilbert action is an analog of the Cartan spin connection equation, and the other can be presented in the form similar to the Einstein equation, with Ricci curvature replaced by the new Ricci type tensor. This tensor generally has a complicated form, but is given in the paper explicitly for variations among semi-symmetric~connections.

math.DG↗

arXiv2016

Variations of the total mixed scalar curvature of a distribution

We examine the total mixed scalar curvature of a fixed distribution as a functional of a pseudo-Riemannian metric. We develop variational formulas for quantities of extrinsic geometry of the distribution to find the critical points of this action. Together with the arbitrary variations of the metric, we consider also variations that preserve the volume of the manifold or partially preserve the metric (e.g. on the distribution). For each of those cases we obtain the Euler-Lagrange equation and its several solutions. The examples of critical metrics that we find are related to various fields of geometry such as contact and 3-Sasakian manifolds, geodesic Riemannian flows and codimension-one foliations of interesting geometric properties (e.g. totally umbilical, minimal).

math.DG↗

arXiv2016

The Einstein-Hilbert type action on foliated pseudo-Riemannian manifolds

We develop variation formulas on almost-product (e.g. foliated) pseudo-Riemannian manifolds, and we consider variations of metric preserving orthogonality of the distributions. These formulae are applied to Einstein-Hilbert type actions: the total mixed scalar curvature and the total extrinsic scalar curvature of a distribution. The obtained Euler-Lagrange equations admit a number of solutions, e.g., twisted products, conformal submersions and isoparametric foliations. The paper generalizes recent results about the actions on codimension-one foliations for the case of arbitrary (co)dimension.

math.DG↗