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Kamil Niedzialomski

Publications and source records attributed to Kamil Niedzialomski.

18 recordsLinked to original sources
arXiv2024

Lifts of maps to frame bundles

Let $(M,g)$ be a Riemannian manifold, $L(M)$ be its frame bundle, $O(M)$ its orthonormal frame bundle. For a distribution $D$ on $M$ we define a subbundle $L(D)\subset L(M)$ or $O(D)\subset O(M)$ in a natural way. This allows us to consider a lift $Lφ$ of a map $φ:M\to N$ not necessarily being a local diffeomorphism. More precisely, if $φ:M\to N$ is a submersion, then $Lφ:L(\mathcal{H}^φ)\to L(N)$ or $Lφ:O(\mathcal{H}^φ)\to L(N)$, where $\mathcal{H}^φ$ is a horizontal distribution of $φ$. Equipping $L(M)$ and $L(N)$ with the Mok metrics, we study conformality and harmonicity of lifts $Lφ$.

math.DG↗
arXiv2024

Two notes on Spin(7)-structures

We derive the explicit formula for the intrinsic torsion of a ${\rm Spin}(7)$-structure on a $8$--dimensional Riemannian manifold $M$. Here, the intrinsic torsion is a difference of the minimal ${\rm Spin}(7)$--connection and the Levi-Civita connection. Hence it is a a section of a bundle $T^{\ast}M\otimes\mathfrak{spin}^{\bot}(M)$. The formula relates the intrinsic torsion with the Lee form $θ$ and $Λ^3_{48}$--component $(δΦ)_{48}$ of a codifferential $δΦ$ of the $4$--form defining a given structure. Using the formula obtained, we compute the condition for a ${\rm Spin}(7)$ structure of type $\mathcal{W}_8$ to be (second order) nearly parallel. Moreover, applying the divergence formula obtained by the author for general Riemannian $G$--structure in another article, we rediscover the well known formula for the scalar curvature in terms of norms of $θ$, $(δΦ)_{48}$ and the divergence ${\rm div}θ$. We justify the formula on appropriate examples.

math.DG↗
arXiv2021

A note on invariant description of $SU(2)$-structures in dimension 5

We develop an invariant approach to $SU(2)$--structures on spin $5$--manifolds. We characterize (via spinor approach) the subspaces in the spinor bundle which induce the same group isomorphic to $SU(2)$. Moreover, we show how to induce quaternionic structure on the contact distribution of considered $SU(2)$--structure. We conclude with the invariance of certain components of the covariant derivative $\nablaφ$, where $φ$ is any spinor field defining considered $SU(2)$--structure. This shows, what expected, that (at least some of) the intrinsic torsion modules, can be derived invariantly with the spinorial approach.

math.DG↗
arXiv2021

An integral formula for $G_2$-structures

We apply an integral formula obtained by the author for a general $G$--structure to the case of $G=G_2$. We derive an integral formula relating curvatures and some quadratic invariants of the endomorphism induced by the intrinsic torsion. We interpret the formula in certain special cases.

math.DG↗
arXiv2019

Harmonic $SU(3)$- and $G_2$-structures via spinors

In this note, using the spinorial description of $SU(3)$ and $G_2$-structures obtained recently by other authors, we give necessary and sufficient conditions for harmonicity of above mentioned structures. We describe obtained results on appropriate homogeneous spaces. Here, harmonicity means harmonicity of the unique section induced by a $G$-structure in consideration.

math.DG↗
arXiv2018

Rodin's formula in arbitrary codimension

We extend the Rodin's formula for $p$--modulus of the family of curves in $\mathbb{R}^n$ to arbitrary codimension. The proof relies on the formula for the $p$--modulus of family of level sets of a submersion and an algebraic lemma relating Jacobi matrices of considered maps. We state appropriate examples.

math.CA↗
arXiv2017

Frame bundle approach to generalized minimal submanifolds

We extend the notion of $r$-minimality of a submanifold in arbitrary codimension to $u$-minimality for a multi-index $u\in\mathbb{N}^q$, where $q$ is the codimension. This approach is based on the analysis on the frame bundle of orthonormal frames of the normal bundle to a submanifold and vector bundles associated with this bundle. The notion of $u$-minimality comes from the variation of $σ_u$-symmetric function obtained from the family of shape operators corresponding to all possible bases of the normal bundle. We obtain the variation field, which gives alternative definition of $u$--minimality. Finally, we give some examples of $u$-minimal submanifolds for some choices of $u$.

math.DG↗
arXiv2016

Geometry of $G$-Structures via the Intrinsic Torsion

We study the geometry of a $G$-structure $P$ inside the oriented orthonormal frame bundle ${\rm SO}(M)$ over an oriented Riemannian manifold $M$. We assume that $G$ is connected and closed, so the quotient ${\rm SO}(n)/G$, where $n=\dim M$, is a normal homogeneous space and we equip ${\rm SO}(M)$ with the natural Riemannian structure induced from the structure on $M$ and the Killing form of ${\rm SO}(n)$. We show, in particular, that minimality of $P$ is equivalent to harmonicity of an induced section of the homogeneous bundle ${\rm SO}(M)\times_{{\rm SO}(n)}{\rm SO}(n)/G$, with a Riemannian metric on $M$ obtained as the pull-back with respect to this section of the Riemannian metric on the considered associated bundle, and to the minimality of the image of this section. We apply obtained results to the case of almost product structures, i.e., structures induced by plane fields.

math.DG↗
arXiv2016

Examples of minimal $G$-structures induced by the Lee form

We compute the condition of minimality of a G-structure for the Gray-Hervella class $\mathcal{W}_4$ of almost hermitian manifolds and $\mathcal{C}_5$ class of almost contact metric structures. We also consider $\mathcal{C}_4$ class by comparison with the Grey-Hervella class $\mathcal{W}_4$. The common feature is the existence of the Lee form $θ$ representing these structures. We show that these classes contain minimal G-structures. Here, minimality means minimality of a $G$-structure inside oriented orthonormal frame bundle $SO(M)$ of a Riemannian manifold $M$.

math.DG↗
arXiv2014

Stable foliations with respect to Fuglede modulus and level sets of $p$--harmonic functions

We continue the study of the variation of the $p$--modulus of a foliation initiated by the first author. We derive the formula for the second variation which allows to study $p$--stable foliations. We obtain some results concerning codimension one $p$--stable foliations. Moreover, we derive the equation for the critical point of the $p$--modulus functional of a foliation given by the level sets of smooth function. We show the correlation with the $q$--harmonicity. We give some examples. In particular, we show that foliations given by the distance function are critical points of $p$--modulus functional.

math.DG↗
arXiv2014

On the frame bundle adapted to a submanifold

Let $M$ be a submanifold of a Riemannian manifold $(N,g)$. $M$ induces a subbundle $O(M,N)$ of adapted frames over $M$ of the bundle of orthonormal frames $O(N)$. Riemannian metric $g$ induces natural metric on $O(N)$. We study the geometry of a submanifold $O(M,N)$ in $O(N)$. We characterize the horizontal distribution of $O(M,N)$ and state its correspondence with the horizontal lift in $O(N)$ induced by the Levi--Civita connection on $N$. In the case of extrinsic geometry, we show that minimality is equivalent to harmonicity of the Gauss map of the submanifold $M$ with deformed Riemannian metric. In the case of intrinsic geometry we compute the curvatures.

math.DG↗
arXiv2012

Two notes on harmonic distributions

We say that a distribution is harmonic if it is harmonic when considered as a section of a Grassmann bundle. We find new examples of harmonic distributions and show nonexistense of harmonic distrubutions on some Riemannian manifolds by two different approaches. Firstly, we lift distributions to the second tangent bundle equipped with the Sasaki metric. Secondly, we deform conformally the metric on a base manifold.

math.DG↗
arXiv2012

On geometry of frame bundles

Let $(M,g)$ be a Riemannian manifold, $L(M)$ its frame bundle. We construct new examples of Riemannian metrics on $L(M)$, which are obtained from Riemannian metrics on the tangent bundle $TM$. We compute the Levi--Civita connection and curvatures of these metrics.

math.DG↗
arXiv2008

On leafwise conformal diffeomorphisms

For every diffeomorphism $φ:M\to N$ between 3--dimensional Riemannian manifolds $M$ and $N$ there are in general locally two 2--dimensional distributions $D_{\pm}$ such that $φ$ is conformal on both of them. We state necessary and sufficient conditions for a distribution to be one of $D_{\pm}$. These are algebraic conditions expressed in terms of the self-adjoint and positive definite operator $(φ_{\ast})^*φ_{\ast}$. We investigate integrability condition of $D_+$ and $D_-$. We also show that it is possible to choose coordinate systems in which leafwise conformal diffeomorphism is holomorphic on leaves.

math.DG↗