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Piccione Paolo - One of the best experts on this subject based on the ideXlab platform.
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Multiplicity of solutions to the Yamabe problem on collapsing Riemannian Submersions
'Mathematical Sciences Publishers', 2013Co-Authors: Bettiol, Renato G., Piccione PaoloAbstract:Let g_t be a family of constant scalar curvature metrics on the total space of a Riemannian Submersion obtained by shrinking the fibers of an original metric g, so that the Submersion collapses as t approaches 0 (i.e., the total space converges to the base in the Gromov-Hausdorff sense). We prove that, under certain conditions, there are at least 3 unit volume constant scalar curvature metrics in the conformal class [g_t] for infinitely many t's accumulating at 0. This holds, e.g., for homogeneous metrics g_t obtained via Cheeger deformation of homogeneous fibrations with fibers of positive scalar curvature.Comment: LaTeX2e, 17 pages. To appear in Pacific J. Mat
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Riemannian Submersions with Discrete Spectrum
SPRINGER, 2012Co-Authors: Bessa G. Pacelli, Montenegro J. Fabio, Piccione PaoloAbstract:We prove some estimates on the spectrum of the Laplacian of the total space of a Riemannian Submersion in terms of the spectrum of the Laplacian of the base and the geometry of the fibers. When the fibers of the Submersions are compact and minimal, we prove that the spectrum of the Laplacian of the total space is discrete if and only if the spectrum of the Laplacian of the base is discrete. When the fibers are not minimal, we prove a discreteness criterion for the total space in terms of the relative growth of the mean curvature of the fibers and the mean curvature of the geodesic spheres in the base. We discuss in particular the case of warped products
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Riemannian Submersions with Discrete Spectrum
NEW YORK, 2012Co-Authors: Bessa G. Pacelli, Montenegro J. Fabio, Piccione PaoloAbstract:We prove some estimates on the spectrum of the Laplacian of the total space of a Riemannian Submersion in terms of the spectrum of the Laplacian of the base and the geometry of the fibers. When the fibers of the Submersions are compact and minimal, we prove that the spectrum of the Laplacian of the total space is discrete if and only if the spectrum of the Laplacian of the base is discrete. When the fibers are not minimal, we prove a discreteness criterion for the total space in terms of the relative growth of the mean curvature of the fibers and the mean curvature of the geodesic spheres in the base. We discuss in particular the case of warped products.CNPq-CAPES (Brazil)CNPqCAPES (Brazil)MEC (Spain)MEC (Spain) [PCI2006-A7-0532
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Maslov index in semi-Riemannian Submersions
'Springer Science and Business Media LLC', 2010Co-Authors: Caponio Erasmo, Javaloyes, Miguel Angel, Piccione PaoloAbstract:We study focal points and Maslov index of a horizontal geodesic $\gamma:I\to M$ in the total space of a semi-Riemannian Submersion $\pi:M\to B$ by determining an explicit relation with the corresponding objects along the projected geodesic $\pi\circ\gamma:I\to B$ in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary space-time which is orthogonal to a timelike Killing vector field.Comment: 20 pages. Title changed, references update
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Maslov index in semi-Riemannian Submersions
SPRINGER, 2010Co-Authors: Caponio Erasmo, Javaloyes, Miguel Angel, Piccione PaoloAbstract:We study focal points and Maslov index of a horizontal geodesic gamma : I -> M in the total space of a semi-Riemannian Submersion pi : M -> B by determining an explicit relation with the corresponding objects along the projected geodesic pi omicron gamma : I -> B in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary spacetime which is orthogonal to a timelike Killing vector field.M.I.U.R.[PRIN07]M.I.U.R.Regional Junta AndaluciaRegional Junta Andalucia[P06-FQM-01951]Fundacion Seneca[04540/GERM/06]Fundacion SenecaSpanish MEC[MTM2009-10418]Spanish MECCapes, Brazil[BEX 1509/08-0]Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES
Luis Guijarro - One of the best experts on this subject based on the ideXlab platform.
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On the metric structure of open manifolds with nonnegative curvature
2015Co-Authors: Luis GuijarroAbstract:Abstract. An open manifold M with nonnegative sectional cur-vature contains a compact totally geodesic submanifold S called the soul. In his solution of the Cheeger–Gromoll conjecture, G. Perelman showed that the metric projection pi:M → S was a C1 Riemannian Submersion which coincided with a map previously constructed by V. Sharafutdinov. In this paper we improve Perelman’s result to show that pi is in fact C2, thus allowing us the use of O’Neill formulas in the study of M. For the proof, we study carefully how the conjugate locus of S behaves in regard to the fibers of pi. As applications, we study souls with totally geodesic Bieberbach submanifolds, and also obtain some rigidity results concerning the distribution of the rays of M. 1. Introduction an
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On the metric structure of open manifolds with nonnegative curvature, preprint
2015Co-Authors: Luis GuijarroAbstract:An open manifold M with nonnegative sectional curvature contains a compact totally geodesic submanifold S called the soul. In his solution of the Cheeger-Gromoll conjecture, G. Perelman showed that the metric projection pi: M → S was a C1 Riemannian Submersion which coincided with a map previously constructed by V. Sharafutdinov. In this paper we improve Perelman’s result to show that pi is in fact C2, thus allowing us the use of O’Neill formulas in the study of M. For the proof, we study carefully how the conjugate locus of S behaves in regard to the fibers of pi. As applications, we study souls with totally geodesic Bieberbach submanifolds, and also obtain some rigidity results concerning the distribution of the rays of M. 1. Introduction an
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when is a Riemannian Submersion homogeneous
Geometriae Dedicata, 2007Co-Authors: Luis Guijarro, Gerard WalschapAbstract:We study the structure of the most common type of Riemannian Submersions, namely those whose fibers are given by the orbits of an isometric group action on a Riemannian manifold. Special emphasis is given to the case where the ambient space has nonnegative curvature.
Vandewalle Stefan - One of the best experts on this subject based on the ideXlab platform.
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A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank
Academic Press, 2013Co-Authors: Vandereycken Bart, Absil P.-a., Vandewalle StefanAbstract:We present a homogeneous space geometry for the manifold of symmetric positive semidefinite matrices of fixed rank. The total space is a connected part of the general linear group endowed with its natural right-invariant metric and the metric on the homogeneous space is chosen such that the quotient space is the image of a Riemannian Submersion from the total space. As a result, we obtain complete geodesics that are the image of certain geodesics on the general linear group. We derive in addition an efficient closed-form expression for these geodesics. Furthermore, we isometrically embed the abstract quotient space into the space of real matrices. This allows us to interpret the vector fields, the metric and the geodesics in terms of concrete matrices. Finally, we compare our geometry with some other geometries in the literature. In particular, we show how other geodesics and their approximations relate to ours.}{homogeneous space; complete geodesic; positive semidefinite cone; low rank matrix; quotient manifold; embedded submanifold.status: publishe
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A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank
Department of Computer Science K.U.Leuven, 2010Co-Authors: Vandereycken Bart, Absil Pierre-antoine, Vandewalle StefanAbstract:We present a homogeneous space geometry for the manifold of symmetric positive semidefinite matrices of fixed rank. The total space is the general linear group endowed with its natural right-invariant metric, and the metric on the homogeneous space is chosen such that the quotient space is the image of a Riemannian Submersion from the total space. As a result, we obtain complete geodesics that are the image of certain geodesics on the general linear group. We derive in addition an efficient closed-form expression for these geodesics. Furthermore, we isometrically embed the abstract quotient space into the space of real matrices. This allows us to interpret the vector fields, the metric and the geodesics in terms of concrete matrices. Finally, we compare our geometry with some other geometries in the literature. In particular, we show how other geodesics and their approximations relate to ours.nrpages: 29status: publishe
Yilmaz Gunduzalp - One of the best experts on this subject based on the ideXlab platform.
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anti invariant Riemannian Submersions from almost product Riemannian manifolds
Mathematical Sciences and Applications E-Notes, 2013Co-Authors: Yilmaz GunduzalpAbstract:In this paper, we introduce anti-invariant Riemannian Submersions from almost product Riemannian manifolds onto Riemannian manifolds.We give an example, investigate the geometry of foliations which are arisenfrom the definition of a Riemannian Submersion and check the harmonicity ofsuch Submersions
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Anti-Invariant Semi-Riemannian Submersions from Almost Para-Hermitian Manifolds
Hindawi Limited, 2013Co-Authors: Yilmaz GunduzalpAbstract:We introduce anti-invariant semi-Riemannian Submersions from almost para-Hermitian manifolds onto semi-Riemannian manifolds. We give an example, investigate the geometry of foliations which are arisen from the definition of a semi-Riemannian Submersion, and check the harmonicity of such Submersions. We also obtain curvature relations between the base manifold and the total manifold
Pro Curtis - One of the best experts on this subject based on the ideXlab platform.
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On Riemannian Submersions and Diffeomorphism Stability
eScholarship University of California, 2012Co-Authors: Pro CurtisAbstract:This thesis consists of work that was carried out in three separate papers that were written during my time at UC, Riverside. Abstract of chapter II: If $\pi:M\rightarrow B$ is a Riemannian Submersion and $M$ has non-negative sectional curvature, O'Neill's Horizontal Curvature Equation shows that $B$ must also have non-negative curvature. We find constraints on the extent to which O'Neill's horizontal curvature equation can be used to create positive curvature on the base space of a Riemannian Submersion. In particular, we study when K. Tapp's theorem on Riemannian Submersions of compact Lie groups with bi-invariant metrics generalizes to arbitrary manifolds of non-negative curvature.Abstract of Chapter III: Though Riemannian Submersions preserve non-negative sectional curvature this does not generalize to Riemannian Submersions from manifolds with non-negative Ricci curvature. We give here an example of a Riemannian Submersion $\pi: M\rightarrow B$ for which $\textrm{Ricci}_p(M)>0$ and at some point $p\in B$, $\text{Ricci}_p(B)<0$. Abstract of Chapter IV: The smallest $r$ so that a metric $r$--ball covers a metric space $M$ is called the radius of $M.$ The volume of a metric $r$-ball in the space form of constant curvature $k$ is an upper bound for the volume of any Riemannian manifold with sectional curvature $\geq k$ and radius $\leq r$. We show that when such a manifold has volume almost equal to this upper bound, it is diffeomorphic to a sphere or a real projective space
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Flats and Submersions in Non-Negative Curvature
2012Co-Authors: Pro Curtis, Wilhelm FrederickAbstract:We find constraints on the extent to which O'Neill's horizontal curvature equation can be used to create positive curvature on the base space of a Riemannian Submersion. In particular, we study when K. Tapp's theorem on Riemannian Submersions of compact Lie groups with bi-invariant metrics generalizes to arbitrary manifolds of non-negative curvature.Comment: 10 page
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Riemannian Submersions Need Not Preserve Positive Ricci Curvature
2012Co-Authors: Pro Curtis, Wilhelm FrederickAbstract:If $\pi :M\rightarrow B$ is a Riemannian Submersion and $M$ has positive sectional curvature, O'Neill's Horizontal Curvature Equation shows that $B$ must also have positive curvature. We show there are Riemannian Submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of (arbitrarily) negative Ricci curvature, but that there are no Riemannian Submersions from manifolds with positive Ricci curvature to manifolds with nonpositive Ricci curvature
