Collapse of warped submersions

Szymon M. Walczak

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Displaying similar documents to “Collapse of warped submersions”

On $G$ -sets and isospectrality

Ori Parzanchevski (2013)

Annales de l’institut Fourier

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We study finite $G$ -sets and their tensor product with Riemannian manifolds, and obtain results on isospectral quotients and covers. In particular, we show the following: If $M$ is a compact connected Riemannian manifold (or orbifold) whose fundamental group has a finite non-cyclic quotient, then $M$ has isospectral non-isometric covers.

The Killing Tensors on an $n$ -dimensional Manifold with $S L (n,)$ -structure

Sergey E. Stepanov, Irina I. Tsyganok, Marina B. Khripunova (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In this paper we solve the problem of finding integrals of equations determining the Killing tensors on an $n$ -dimensional differentiable manifold $M$ endowed with an equiaffine $S L (n,)$ -structure and discuss possible applications of obtained results in Riemannian geometry.

The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds

Jan Kurek, Włodzimierz Mikulski (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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If $(M, g)$ is a Riemannian manifold, we have the well-known base preserving vector bundle isomorphism $T M \tilde{=} T^{*} M$ given by $v \to g (v, -)$ between the tangent $T M$ and the cotangent $T^{*} M$ bundles of $M$ . In the present note, we generalize this isomorphism to the one $T^{(r)} M \tilde{=} T^{r *} M$ between the $r$ -th order vector tangent $T^{(r)} M = {(J^{r} {(M, R)}_{0})}^{*}$ and the $r$ -th order cotangent $T^{r *} M = J^{r} {(M, R)}_{0}$ bundles of $M$ . Next, we describe all base preserving vector bundle maps $C_{M} (g) : T^{(r)} M \to T^{r *} M$ depending on a Riemannian metric $g$ in terms of natural (in $g$ ) tensor fields on $M$ .

Generalized gradient flow and singularities of the Riemannian distance function

Piermarco Cannarsa (2012-2013)

Séminaire Laurent Schwartz — EDP et applications

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Significant information about the topology of a bounded domain $Ω$ of a Riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partial Ω}$ , from the boundary of $Ω$ . We discuss recent results showing the invariance of the singular set of the distance function with respect to the generalized gradient flow of $d_{\partial Ω}$ , as well as applications to homotopy equivalence.

$η$ -Ricci Solitons on $η$ -Einstein ${(L C S)}_{n}$ -Manifolds

Shyamal Kumar Hui, Debabrata Chakraborty (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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The object of the present paper is to study $η$ -Ricci solitons on $η$ -Einstein ${(L C S)}_{n}$ -manifolds. It is shown that if $ξ$ is a recurrent torse forming $η$ -Ricci soliton on an $η$ -Einstein ${(L C S)}_{n}$ -manifold then $ξ$ is (i) concurrent and (ii) Killing vector field.

Deformations of Metrics and Biharmonic Maps

Aicha Benkartab, Ahmed Mohammed Cherif (2020)

Communications in Mathematics

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We construct biharmonic non-harmonic maps between Riemannian manifolds $(M, g)$ and $(N, h)$ by first making the ansatz that $ϕ : (M, g) \to (N, h)$ be a harmonic map and then deforming the metric on $N$ by ${\tilde{h}}_{α} = α h + (1 - α) d f \otimes d f$ to render $ϕ$ biharmonic, where $f$ is a smooth function with gradient of constant norm on $(N, h)$ and $α \in (0, 1)$ . We construct new examples of biharmonic non-harmonic maps, and we characterize the biharmonicity of some curves on Riemannian manifolds.

Complete Riemannian manifolds admitting a pair of Einstein-Weyl structures

Amalendu Ghosh (2016)

Mathematica Bohemica

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We prove that a connected Riemannian manifold admitting a pair of non-trivial Einstein-Weyl structures $(g, \pm ω)$ with constant scalar curvature is either Einstein, or the dual field of $ω$ is Killing. Next, let $(M^{n}, g)$ be a complete and connected Riemannian manifold of dimension at least $3$ admitting a pair of Einstein-Weyl structures $(g, \pm ω)$ . Then the Einstein-Weyl vector field $E$ (dual to the $1$ -form $ω$ ) generates an infinitesimal harmonic transformation if and only if $E$ is Killing.

A compactness result for polyharmonic maps in the critical dimension

Shenzhou Zheng (2016)

Czechoslovak Mathematical Journal

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For $n = 2 m \geq 4$ , let $Ω \in ℝ^{n}$ be a bounded smooth domain and $𝒩 \subset ℝ^{L}$ a compact smooth Riemannian manifold without boundary. Suppose that ${u_{k}} \in W^{m, 2} (Ω, 𝒩)$ is a sequence of weak solutions in the critical dimension to the perturbed $m$ -polyharmonic maps $\frac{d}{d t} |_{t = 0} E_{m} (Π (u + t ξ)) = 0$ with $Φ_{k} \to 0$ in ${(W^{m, 2} (Ω, 𝒩))}^{*}$ and $u_{k} ⇀ u$ weakly in $W^{m, 2} (Ω, 𝒩)$ . Then $u$ is an $m$ -polyharmonic map. In particular, the space of $m$ -polyharmonic maps is sequentially compact for the weak- $W^{m, 2}$ topology.

On the classification of $3$ -dimensional $F$ -manifold algebras

Zhiqi Chen, Jifu Li, Ming Ding (2022)

Czechoslovak Mathematical Journal

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$F$ -manifold algebras are focused on the algebraic properties of the tangent sheaf of $F$ -manifolds. The local classification of 3-dimensional $F$ -manifolds has been given in A. Basalaev, C. Hertling (2021). We study the classification of 3-dimensional $F$ -manifold algebras over the complex field $ℂ$ .

On the multiplicity of eigenvalues of conformally covariant operators

Yaiza Canzani (2014)

Annales de l’institut Fourier

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Let $(M, g)$ be a compact Riemannian manifold and $P_{g}$ an elliptic, formally self-adjoint, conformally covariant operator of order $m$ acting on smooth sections of a bundle over $M$ . We prove that if $P_{g}$ has no rigid eigenspaces (see Definition 2.2), the set of functions $f \in C^{\infty} (M, ℝ)$ for which $P_{e^{f} g}$ has only simple non-zero eigenvalues is a residual set in $C^{\infty} (M, ℝ)$ . As a consequence we prove that if $P_{g}$ has no rigid eigenspaces for a dense set of metrics, then all non-zero eigenvalues are simple for a residual set of metrics...

Growth of a primitive of a differential form

Jean-Claude Sikorav (2001)

Bulletin de la Société Mathématique de France

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For an exact differential form on a Riemannian manifold to have a primitive bounded by a given function $f$ , by Stokes it has to satisfy some weighted isoperimetric inequality. We show the converse up to some constants if $M$ has bounded geometry. For a volume form, it suffices to have the inequality ( $| Ω | \leq \int_{\partial Ω} f d σ$ for every compact domain $Ω \subset M$ ). This implies in particular the “well-known” result that if $M$ is the universal covering of a compact Riemannian manifold with non-amenable fundamental group, then...

Holomorphic line bundles and divisors on a domain of a Stein manifold

Makoto Abe (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let $D$ be an open set of a Stein manifold $X$ of dimension $n$ such that $H^{k} (D, 𝒪) = 0$ for $2 \leq k \leq n - 1$ . We prove that $D$ is Stein if and only if every topologically trivial holomorphic line bundle $L$ on $D$ is associated to some Cartier divisor $𝔡$ on $D$ .

A characterization of the Riemann extension in terms of harmonicity

Cornelia-Livia Bejan, Şemsi Eken (2017)

Czechoslovak Mathematical Journal

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If $(M, \nabla)$ is a manifold with a symmetric linear connection, then $T^{*} M$ can be endowed with the natural Riemann extension $\bar{g}$ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to $\bar{g}$ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure $𝒫$ on $(T^{*} M, \bar{g})$ and prove that $𝒫$ is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if $\bar{g}$ reduces...