The Kasparov product on submersions of open manifolds

2020 ◽  
pp. 1-35
Author(s):  
Koen van den Dungen
Keyword(s):  
Elliptic Operator ◽  
Riemannian Manifolds ◽  
Riemannian Submersion ◽  
Base Manifold ◽  
Fundamental Class ◽  
Open Manifolds ◽  
Incomplete Riemannian Manifolds ◽  
Symmetric Operators ◽  
Regular Operators ◽  
Construction Works

We study the Kasparov product on (possibly non-compact and incomplete) Riemannian manifolds. Specifically, we show on a submersion of Riemannian manifolds that the tensor sum of a regular vertically elliptic operator on the total space and an elliptic operator on the base space represents the Kasparov product of the corresponding classes in [Formula: see text]-theory. This construction works in general for symmetric operators (i.e. without assuming self-adjointness), and extends known results for submersions with compact fibers. The assumption of regularity for the vertically elliptic operator is not always satisfied, but depends on the topology and geometry of the submersion, and we give explicit examples of non-regular operators. We apply our main result to obtain a factorization in unbounded [Formula: see text]-theory of the fundamental class of a Riemannian submersion, as a Kasparov product of the shriek map of the submersion and the fundamental class of the base manifold.

scholarly journals Anti-Invariant Semi-Riemannian Submersions from Almost Para-Hermitian Manifolds

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Yılmaz Gündüzalp
Keyword(s):  
Riemannian Manifolds ◽  
Riemannian Submersion ◽  
Riemannian Submersions ◽  
Base Manifold ◽  
Hermitian Manifolds ◽  
Definition Of

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.

Anti-invariant Riemannian submersions from Sasakian manifolds with totally umbilical fibers

2021 ◽  
pp. 2150169
Author(s):  
Gizem Köprülü ◽  
Bayram Şahin
Keyword(s):  
Vector Field ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Sasakian Manifold ◽  
Riemannian Submersion ◽  
Sasakian Manifolds ◽  
Riemannian Submersions ◽  
Totally Umbilical ◽  
Characteristic Vector Field ◽  
Horizontal Vector

The purpose of this paper is to study anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds such that characteristic vector field is vertical or horizontal vector field. We first show that any anti-invariant Riemannian submersions from Sasakian manifold is not a Riemannian submersion with totally umbilical fiber. Then we introduce anti-invariant Riemannian submersions from Sasakian manifolds with totally contact umbilical fibers. We investigate the totally contact geodesicity of fibers of such submersions. Moreover, under this condition, we investigate Ricci curvature of anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds.

scholarly journals Anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds

Filomat ◽  
2015 ◽  
Vol 29 (7) ◽  
pp. 1429-1444 ◽  
Author(s):  
Cengizhan Murathan ◽  
Erken Küpeli
Keyword(s):  
Riemannian Manifolds ◽  
Riemannian Submersion ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Riemannian Submersions ◽  
Totally Geodesic ◽  
Cosymplectic Manifolds ◽  
Necessary And Sufficient ◽  
Decomposition Theorems ◽  
Characteristic Vector Field

We introduce anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We survey main results of anti-invariant Riemannian submersions defined on cosymplectic manifolds. We investigate necessary and sufficient condition for an anti-invariant Riemannian submersion to be totally geodesic and harmonic. We give examples of anti-invariant submersions such that characteristic vector field ? is vertical or horizontal. Moreover we give decomposition theorems by using the existence of anti-invariant Riemannian submersions.

scholarly journals A note on warped product almost quasi-Yamabe solitons

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 2009-2016 ◽  
Author(s):  
Adara Blaga
Keyword(s):  
Riemannian Manifolds ◽  
Warped Product ◽  
Sufficient Conditions ◽  
Constant Scalar Curvature ◽  
Product Manifold ◽  
Base Manifold ◽  
Warped Product Manifold ◽  
Necessary And Sufficient ◽  
Yamabe Solitons ◽  
Yamabe Soliton

We consider almost quasi-Yamabe solitons in Riemannian manifolds, derive a Bochner-type formula in the gradient case and prove that under certain assumptions, the manifold is of constant scalar curvature. We also provide necessary and sufficient conditions for a gradient almost quasi-Yamabe soliton on the base manifold to induce a gradient almost quasi-Yamabe soliton on the warped product manifold.

scholarly journals Hemi-slant ξ⊥-Riemannian submersions in contact geometry

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3747-3758
Author(s):  
Ramazan Sari ◽  
Mehmet Akyol
Keyword(s):  
Riemannian Manifolds ◽  
Natural Generalization ◽  
Product Manifold ◽  
Necessary And Sufficient Condition ◽  
Sasakian Manifolds ◽  
Riemannian Submersions ◽  
Space Forms ◽  
Base Manifold ◽  
Sasakian Space Forms ◽  
Necessary And Sufficient

M. A. Akyol and R. Sar? [On semi-slant ??-Riemannian submersions, Mediterr. J. Math. 14(6) (2017) 234.] defined semi-slant ??-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. As a generalization of the above notion and natural generalization of anti-invariant ??-Riemannian submersions, semi-invariant ??-Riemannian submersions and slant submersions, we study hemi-slant ??-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We obtain the geometry of foliations, give some examples and find necessary and sufficient condition for the base manifold to be a locally product manifold. Moreover, we obtain some curvature relations from Sasakian space forms between the total space, the base space and the fibres.

scholarly journals REILLY INEQUALITIES OF ELLIPTIC OPERATORS ON CLOSED SUBMANIFOLDS

2009 ◽  
Vol 80 (2) ◽  
pp. 335-346
Author(s):  
RUSHAN WANG
Keyword(s):  
Elliptic Operator ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Upper Bound ◽  
Vector Fields ◽  
Elliptic Operators ◽  
Position Vector ◽  
Space Forms ◽  
Nonzero Eigenvalue ◽  
Closed Hypersurfaces

AbstractUsing generalized position vector fields we obtain new upper bound estimates of the first nonzero eigenvalue of a kind of elliptic operator on closed submanifolds isometrically immersed in Riemannian manifolds of bounded sectional curvature. Applying these Reilly inequalities we improve a series of recent upper bound estimates of the first nonzero eigenvalue of the Lr operator on closed hypersurfaces in space forms.

scholarly journals Estimates for Eigenvalues of the Elliptic Operator in Divergence Form on Riemannian Manifolds

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Shenyang Tan ◽  
Tiren Huang ◽  
Wenbin Zhang
Keyword(s):  
Elliptic Operator ◽  
Riemannian Manifolds ◽  
Special Functions ◽  
Type Inequality ◽  
Divergence Form ◽  
Hadamard Manifolds ◽  
Complete Submanifolds ◽  
Universal Inequalities ◽  
Complete Manifolds ◽  
Bounded Below

We investigate the Dirichlet weighted eigenvalue problem of the elliptic operator in divergence form on compact Riemannian manifolds(M,g,e-ϕdv). We establish a Yang-type inequality of this problem. We also get universal inequalities for eigenvalues of elliptic operators in divergence form on compact domains of complete submanifolds admitting special functions which include the Hadamard manifolds with Ricci curvature bounded below and any complete manifolds admitting eigenmaps to a sphere.

scholarly journals On the L2-Poincaré duality for incomplete Riemannian manifolds: A general construction with applications

2016 ◽  
Vol 08 (01) ◽  
pp. 151-186 ◽  
Author(s):  
Francesco Bei
Keyword(s):  
Riemannian Manifolds ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Finite Dimensional Vector Space ◽  
Dimension Formula ◽  
Finite Dimensional ◽  
Rham Complex ◽  
De Rham ◽  
Hilbert Complex ◽  
Incomplete Riemannian Manifolds

Let [Formula: see text] be an open, oriented and incomplete Riemannian manifold of dimension [Formula: see text]. Under some general conditions we show the existence of a Hilbert complex [Formula: see text] such that its cohomology groups, labeled with [Formula: see text], satisfy the following properties: [Formula: see text] [Formula: see text] (Poincaré duality holds) There exists a well-defined and nondegenerate pairing: [Formula: see text] If [Formula: see text] is a Fredholm complex, then every closed extension of the de Rham complex [Formula: see text] is a Fredholm complex and, for each [Formula: see text], the quotient [Formula: see text] is a finite dimensional vector space.

scholarly journals Semi-invariant Submersions from Almost Hermitian Manifolds

2013 ◽  
Vol 56 (1) ◽  
pp. 173-183 ◽  
Author(s):  
Bayram Ṣahin
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Sufficient Conditions ◽  
Riemannian Submersion ◽  
Riemannian Submersions ◽  
Totally Geodesic ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Definition Of ◽  
Necessary And Sufficient

AbstractWe introduce semi-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations that arise from the definition of a Riemannian submersion, and find necessary sufficient conditions for total manifold to be a locally product Riemannian manifold. We also find necessary and sufficient conditions for a semi-invariant submersion to be totally geodesic. Moreover, we obtain a classification for semiinvariant submersions with totally umbilical fibers and show that such submersions put some restrictions on total manifolds.

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