scholarly journals Osserman pseudo-Riemannian manifolds of signature (2,2)

2001 ◽  
Vol 71 (3) ◽  
pp. 367-396 ◽  
Author(s):  
Novica Blažić ◽  
Neda Bokan ◽  
Zoran Rakić
Keyword(s):  
Riemannian Manifold ◽  
Characteristic Polynomial ◽  
Riemannian Manifolds ◽  
Jacobi Operator ◽  
Tangent Vector ◽  
Jordan Form ◽  
The Other ◽  
Jacobi Operators ◽  
Osserman Manifold ◽  
Rank One

AbstractA pseudo-Riemannian manifold is said to be timelike (spacelike) Osserman if the Jordan form of the Jacobi operator Kx is independent of the particular unit timelike (spacelike) tangent vector X. The first main result is that timelike (spacelike) Osserman manifold (M, g) of signature (2, 2) with the diagonalizable Jacobi operator is either locally rank-one symmetric or flat. In the nondiagonalizable case the characteristic polynomial of Kx has to have a triple zero, which is the other main result. An important step in the proof is based on Walker's study of pseudo-Riemannian manifolds admitting parallel totally isotropic distributions. Also some interesting additional geometric properties of Osserman type manifolds are established. For the nondiagonalizable Jacobi operators some of the examples show a nature of the Osserman condition for Riemannian manifolds different from that of pseudo-Riemannian manifolds.

scholarly journals Transversal Jacobi Operators in Almost Contact Manifolds

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 31
Author(s):  
Jong Taek Cho ◽  
Makoto Kimura
Keyword(s):  
Riemannian Manifold ◽  
Hyperbolic Space ◽  
Jacobi Operator ◽  
Real Hypersurfaces ◽  
Contact Manifolds ◽  
Jacobi Operators ◽  
Ruled Real Hypersurfaces ◽  
Contact Distribution ◽  
Contact Riemannian Manifold

Along a transversal geodesic γ whose tangent belongs to the contact distribution D, we define the transversal Jacobi operator Rγ=R(·,γ˙)γ˙ on an almost contact Riemannian manifold M. Then, using the transversal Jacobi operator Rγ, we give a new characterization of the Sasakian sphere. In the second part, we characterize the complete ruled real hypersurfaces in complex hyperbolic space.

scholarly journals On some types of geodesics on Riemannian manifolds

1981 ◽  
Vol 81 ◽  
pp. 27-43 ◽  
Author(s):  
Tetsunori Kurogi
Keyword(s):  
Riemannian Manifold ◽  
Critical Point ◽  
Riemannian Manifolds ◽  
Critical Point Theory ◽  
Point Theory ◽  
The Other ◽  
Existence Theory ◽  
Infinite Dimensional ◽  
Quantitative Theory

For a given Riemannian manifold M and its submanifold N, one can find various types of geodesies on M starting from any point of N and ending in any point of N. For example, geodesies which start perpendicularly from N and end perpendicularly in N are treated by many mathematicians. K. Grove has stated a condition in a general case for the existence of such a geodesic ([4]), where he has used the method of the infinite dimensional critical point theory. This method is very useful for the study of geodesies and many geometricians have used it successfully. It has two aspects: one is an existence theory and the other is a quantitative theory, which one can find, for instance, in the excellent theory for closed geodesies of W. Klingenberg ([1], [7]) and so on.

scholarly journals Non-invariant submanifolds of locally decomposable golden Riemannian manifolds

2021 ◽  
Author(s):  
Mustafa Gök ◽  
Erol Kılıç
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Tangent Vector ◽  
Invariant Submanifold ◽  
Ambient Manifold ◽  
Invariant Submanifolds ◽  
Induced Structure

AbstractIn this paper, we investigate any non-invariant submanifold of a locally decomposable golden Riemannian manifold in the case that the rank of the set of tangent vector fields of the induced structure on the submanifold by the golden structure of the ambient manifold is less than or equal to the codimension of the submanifold.

scholarly journals $L^{p}$ harmonic 1-forms on conformally flat Riemannian manifolds

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Li ◽  
Shuxiang Feng ◽  
Peibiao Zhao
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Finiteness Theorem ◽  
Conformally Flat ◽  
Locally Conformally Flat ◽  
Flat Riemannian Manifold ◽  
Ricci Form

AbstractIn this paper, we establish a finiteness theorem for $L^{p}$ L p harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han’s result on $L^{2}$ L 2 harmonic 1-forms.

Bézier Curves on Riemannian Manifolds and Lie Groups With Kinematics Applications

1994 ◽  
Author(s):  
Frank C. Park ◽  
Bahram Ravani
Keyword(s):  
Riemannian Manifold ◽  
Rigid Body ◽  
Lie Groups ◽  
Riemannian Manifolds ◽  
Group Structure ◽  
Recursive Algorithm ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Spatial Displacements ◽  
Natural Way

Abstract In this article we generalize the concept of Bézier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau’s algorithm for constructing Bézier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their algebraic group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Bézier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Bezier sense) using this recursive algorithm. We apply this algorithm to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body.

scholarly journals On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Anna Maria Micheletti ◽  
Angela Pistoia
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Elliptic Problem ◽  
Multiplicity Result ◽  
Nonlinear Elliptic Problem ◽  
Nodal Solutions ◽  
Nonlinear Elliptic ◽  
Sign Changing Solutions

Given thatis a smooth compact and symmetric Riemannian -manifold, , we prove a multiplicity result for antisymmetric sign changing solutions of the problem in . Here if and if .

scholarly journals Hyperbolic rank rigidity for manifolds of -pinched negative curvature

2018 ◽  
Vol 40 (5) ◽  
pp. 1194-1216
Author(s):  
CHRIS CONNELL ◽  
THANG NGUYEN ◽  
RALF SPATZIER
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Negative Curvature ◽  
Jacobi Field ◽  
Rigidity Result ◽  
Locally Symmetric Space ◽  
Rank One ◽  
Sectional Curvatures

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac{1}{4}]$ and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

Moon Lake’s Orphans and “The Other Way to Live”

2019 ◽  
pp. 79-102
Author(s):  
Jean C. Griffith
Keyword(s):  
Twentieth Century ◽  
Child Welfare ◽  
Welfare Policy ◽  
The Other ◽  
American Democracy ◽  
Child Welfare Policy ◽  
Early Twentieth Century ◽  
Rank One ◽  
Poor Children ◽  
Moon Lake

This essay examines the roles the character Easter in “Moon Lake” plays in the context of early-twentieth-century debates about the roots of poverty and society’s level of responsibility to poor children. By placing the focus of the story not on Easter but on the genteel Morgana girls’ shifting attitudes about her, Welty illustrates the ways child welfare policy was shaped by conflicting attitudes, whereby sympathy for innocent children coexisted with scorn for their parents. Assuming that Easter lives outside the boundaries that mark their own places in Morgana’s gendered, class-bound, and racially-segregated society, Jinny Love Stark and Nina Carmichael imagine the “orphan” to embody a womanhood untethered by race or rank, one, perhaps, more representative of American democracy. Ultimately, though, the girls come to see that Easter’s status as an orphan makes her more marked by and vulnerable to the violence and oppression that shape the South’s racial patriarchy.

scholarly journals A volume estimate for strong subharmonicity and maximum principle on complete Riemannian manifolds

1998 ◽  
Vol 151 ◽  
pp. 25-36 ◽  
Author(s):  
Kensho Takegoshi
Keyword(s):  
Riemannian Manifold ◽  
Maximum Principle ◽  
Growth Condition ◽  
Riemannian Manifolds ◽  
Volume Growth ◽  
Complete Riemannian Manifold ◽  
Volume Estimate ◽  
Complete Riemannian Manifolds ◽  
Generalized Maximum Principle

Abstract.A generalized maximum principle on a complete Riemannian manifold (M, g) is shown under a certain volume growth condition of (M, g) and its geometric applications are given.

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