scholarly journals SYMPLECTIC CONNECTIONS WITH A PARALLEL RICCI CURVATURE

2003 ◽  
Vol 46 (3) ◽  
pp. 747-766 ◽  
Author(s):  
Charles Boubel
Keyword(s):  
Riemannian Manifold ◽  
Symplectic Manifold ◽  
Ricci Curvature ◽  
Ricci Tensor ◽  
Decomposition Theorem ◽  
Symplectic Connections ◽  
Symplectic Spaces ◽  
Levi Civita Connection ◽  
Parallel Ricci Tensor ◽  
Symplectic Connection

AbstractA symplectic connection on a symplectic manifold, unlike the Levi-Civita connection on a Riemannian manifold, is not unique. However, some spaces admit a canonical connection (symmetric symplectic spaces, Kähler manifolds, etc.); besides, some ‘preferred’ symplectic connections can be defined in some situations. These facts motivate a study of the symplectic connections, inducing a parallel Ricci tensor. This paper gives the possible forms of the Ricci curvature on such manifolds and gives a decomposition theorem (linked with the holonomy decomposition) for them.AMS 2000 Mathematics subject classification: Primary 53B05; 53B30; 53B35; 53C25; 53C55

Real Hypersurfaces in Complex Two-plane Grassmannians with Reeb Parallel Ricci Tensor in the GTW Connection

2016 ◽  
Vol 59 (4) ◽  
pp. 721-733
Author(s):  
Juan de Dios Pérez ◽  
Hyunjin Lee ◽  
Young Jin Suh ◽  
Changhwa Woo
Keyword(s):  
Ricci Tensor ◽  
Complete Classification ◽  
Real Hypersurfaces ◽  
Classification Problems ◽  
Hopf Hypersurfaces ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Parallel Ricci Tensor

AbstractThere are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians G2(ℂm+2). Among them, Suh classified Hopf hypersurfaces in G2(ℂm+2) with Reeb parallel Ricci tensor in Levi–Civita connection. In this paper, we introduce the notion of generalized Tanaka–Webster (GTW) Reeb parallel Ricci tensor for Hopf hypersurfaces in G2(ℂm+2). Next, we give a complete classification of Hopf hypersurfaces in G2(ℂm+2) with GTW Reeb parallel Ricci tensor.

scholarly journals Curvature of quaternionic Kähler manifolds with $$S^1$$-symmetry

2021 ◽  
Author(s):  
V. Cortés ◽  
A. Saha ◽  
D. Thung
Keyword(s):  
Curvature Tensor ◽  
Decomposition Theorem ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Cohomogeneity One ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Quaternionic Kähler

AbstractWe study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic Kähler side in terms of the initial hyper-Kähler data. Our curvature formula refines a well-known decomposition theorem due to Alekseevsky. As an application, we compute the norm of the curvature tensor for a series of complete quaternionic Kähler manifolds arising from flat hyper-Kähler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.

HARNACK-TYPE INEQUALITIES FOR THE POROUS MEDIUM EQUATION ON A MANIFOLD WITH NON-NEGATIVE RICCI CURVATURE

2012 ◽  
Vol 23 (04) ◽  
pp. 1250009 ◽  
Author(s):  
JEONGWOOK CHANG ◽  
JINHO LEE
Keyword(s):  
Porous Medium ◽  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Porous Medium Equation ◽  
Complete Riemannian Manifold ◽  
Gradient Estimates ◽  
Medium Equation

We derive Harnack-type inequalities for non-negative solutions of the porous medium equation on a complete Riemannian manifold with non-negative Ricci curvature. Along with gradient estimates, reparametrization of a geodesic and time rescaling of a solution are key tools to get the results.

On Conformally Flat Spaces with Commuting Curvature and Ricci Transformations

1972 ◽  
Vol 24 (5) ◽  
pp. 799-804 ◽  
Author(s):  
R. L. Bishop ◽  
S.I. Goldberg
Keyword(s):  
Riemannian Manifold ◽  
Covariant Derivative ◽  
Ricci Tensor ◽  
Vector Fields ◽  
Conformally Flat ◽  
Image Position

Let (M, g) be a C∞ Riemannian manifold and A be the field of symmetric endomorphisms corresponding to the Ricci tensor S; that is,We consider a condition weaker than the requirement that A be parallel (▽ A = 0), namely, that the “second exterior covariant derivative” vanish ( ▽x▽YA — ▽Y ▽XA — ▽[X,Y]A = 0), which by the classical interchange formula reduces to the propertywhere R(X, Y) is the curvature transformation determined by the vector fields X and Y.The property (P) is equivalent toTo see this we observe first that a skew symmetric and a symmetric endomorphism commute if and only if their product is skew symmetric.

scholarly journals Volume comparison of Bishop-Gromov type

1992 ◽  
Vol 45 (2) ◽  
pp. 241-248
Author(s):  
Sungyun Lee
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Complete Riemannian Manifold ◽  
Comparison Theorems ◽  
Volume Comparison

Bishop-Gromov type comparison theorems for the volume of a tube about a sub-manifold of a complete Riemannian manifold whose Ricci curvature is bounded from below are proved. The Kähler analogue is also proved.

2D RICCI FLAT GRADIENT SOLITONS ARISING FROM REMARKABLE MODELS IN PHYSICS

2013 ◽  
Vol 10 (10) ◽  
pp. 1350059
Author(s):  
GABRIEL BERCU
Keyword(s):  
Riemannian Manifold ◽  
Local Computation ◽  
Ricci Flat ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Gradient Solitons

On a pseudo-Riemannian manifold (M, g) we consider ∇ the Levi-Cività connection associated to metric g and a function f : M → ℝ whose pseudo-Riemannian Hessian [Formula: see text] is non-degenerate and with constant signature. We study properties of the pseudo-Riemannian manifold (M, h), [Formula: see text] in terms of local computation. Investigating the conditions for existence of the equal connections [Formula: see text], produced by g and h, we determine classes of explicit Ricci flat gradient solitons for some particular forms of g, arising from remarkable physics models.

scholarly journals Gaussian heat kernel estimates: From functions to forms

2020 ◽  
Vol 2020 (761) ◽  
pp. 25-79
Author(s):  
Thierry Coulhon ◽  
Baptiste Devyver ◽  
Adam Sikora
Keyword(s):  
Riemannian Manifold ◽  
Heat Kernel ◽  
Ricci Curvature ◽  
Compact Riemannian Manifold ◽  
Kernel Estimates ◽  
Doubling Property ◽  
Negative Part ◽  
Heat Kernel Estimates ◽  
Volume Doubling ◽  
Upper Estimate

AbstractOn a complete non-compact Riemannian manifold satisfying the volume doubling property, we give conditions on the negative part of the Ricci curvature that ensure that, unless there are harmonic 1-forms, the Gaussian heat kernel upper estimate on functions transfers to one-forms. These conditions do no entail any constraint on the size of the Ricci curvature, only on its decay at infinity.

scholarly journals Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

2019 ◽  
Author(s):  
Debora Impera ◽  
Michele Rimoldi ◽  
Giona Veronelli
Keyword(s):  
Maximum Principle ◽  
Ricci Curvature ◽  
Sobolev Spaces ◽  
Upper Bound ◽  
Ricci Tensor ◽  
Injectivity Radius ◽  
Quadratic Growth ◽  
Riemann Curvature ◽  
Main Application ◽  
Compactly Supported Functions

Abstract We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in $W^{2,p}$. The result is improved for $p=2$ avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications we prove new disturbed Sobolev and Calderón–Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori–Yau maximum principle for the Hessian.

scholarly journals On inextensible flows of tangent developable of biharmonic B-Slant helices according to Bishop frames in the special 3-dimensional Kenmotsu manifold

2011 ◽  
Vol 31 (1) ◽  
pp. 89 ◽  
Author(s):  
Vedat Asil ◽  
Talat Körpınar ◽  
Essin Turhan
Keyword(s):  
Ricci Tensor ◽  
Three Dimensional ◽  
Kenmotsu Manifold ◽  
3 Dimensional ◽  
Developable Surfaces ◽  
Tangent Developable ◽  
Inextensible Flows ◽  
Parallel Ricci Tensor

In this paper, we study inextensible flows of tangent developable surfaces of biharmonic B-slant helices in the special three-dimensional Kenmotsu manifold K with η-parallel ricci tensor. We express some interesting relations about inextensible flows of this surfaces.

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