Spherical maps and three-dimensional torsion of surfaces in four-dimensional Riemannian manifolds. II

1991 ◽  
Vol 31 (5) ◽  
pp. 756-762
Author(s):  
S. E. Kozlov
Keyword(s):  
Riemannian Manifolds ◽  
Three Dimensional ◽  
Spherical Maps

scholarly journals Alternative Approach to Infinity

1974 ◽  
Vol 64 ◽  
pp. 99-99
Author(s):  
Peter G. Bergmann
Keyword(s):  
Large Class ◽  
Riemannian Manifolds ◽  
Quotient Space ◽  
Three Dimensional ◽  
Cauchy Data ◽  
Space Time ◽  
Equivalence Classes ◽  
Null Infinity ◽  
Alternative Approach ◽  
Dimensional Domain

Following Penrose's construction of space-time infinity by means of a conformal construction, in which null-infinity is a three-dimensional domain, whereas time- and space-infinities are points, Geroch has recently endowed space-infinity with a somewhat richer structure. An approach that might work with a large class of pseudo-Riemannian manifolds is to induce a topology on the set of all geodesics (whether complete or incomplete) by subjecting their Cauchy data to (small) displacements in space-time and Lorentz rotations, and to group the geodesics all of whose neighborhoods intersect into equivalence classes. The quotient space of geodesics over equivalence classes is to represent infinity. In the case of Minkowski, null-infinity has the usual structure, but I0, I+, and I- each become three-dimensional as well.

THE PRIME GEODESIC THEOREM AND QUANTUM MECHANICS ON FINITE VOLUME GRAPHS: A REVIEW

2001 ◽  
Vol 13 (12) ◽  
pp. 1459-1503 ◽  
Author(s):  
NORMAN E. HURT
Keyword(s):  
Finite Volume ◽  
Riemannian Manifolds ◽  
Riemann Surfaces ◽  
Three Dimensional ◽  
Hyperbolic Spaces ◽  
Quantum Graphs ◽  
Selberg Zeta Function ◽  
Ramanujan Graphs ◽  
Prime Geodesic Theorem ◽  
Ramanujan Conjecture

The prime geodesic theorem is reviewed for compact and finite volume Riemann surfaces and for finite and finite volume graphs. The methodology of how these results follow from the theory of the Selberg zeta function and the Selberg trace formula is outlined. Relationships to work on quantum graphs are surveyed. Extensions to compact Riemannian manifolds, in particular to three-dimensional hyperbolic spaces, are noted. Interconnections to the Selberg eigenvalue conjecture, the Ramanujan conjecture and Ramanujan graphs are developed.

scholarly journals Point interactions in two- and three-dimensional Riemannian manifolds

2010 ◽  
Vol 43 (33) ◽  
pp. 335204 ◽  
Author(s):  
Fatih Erman ◽  
O Teoman Turgut
Keyword(s):  
Riemannian Manifolds ◽  
Three Dimensional ◽  
Point Interactions

YAMABE TYPE EQUATIONS ON THREE DIMENSIONAL RIEMANNIAN MANIFOLDS

1999 ◽  
Vol 01 (01) ◽  
pp. 1-50 ◽  
Author(s):  
YANYAN LI ◽  
MEIJUN ZHU
Keyword(s):  
Riemannian Manifolds ◽  
Compact Riemannian Manifold ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Dimensional Sphere ◽  
Compact Set ◽  
Curvature Function ◽  
Necessary And Sufficient Condition ◽  
All Solutions ◽  
Necessary And Sufficient

A theorem of Escobar and Schoen asserts that on a positive three dimensional smooth compact Riemannian manifold which is not conformally equivalent to the standard three dimensional sphere, a necessary and sufficient condition for a C2 function K to be the scalar curvature function of some conformal metric is that K is positive somewhere. We show that for any positive C2 function K, all such metrics stay in a compact set with respect to C3 norms and the total Leray-Schauder degree of all solutions is equal to -1. Such existence and compactness results no longer hold in such generality in higher dimensions or on manifolds conformally equivalent to standard three dimensional spheres. The results are also established for more general Yamabe type equations on three dimensional manifolds.

scholarly journals Bishop and Laplacian Comparison Theorems on Three-Dimensional Contact Sub-Riemannian Manifolds with Symmetry

2013 ◽  
Vol 25 (1) ◽  
pp. 512-535 ◽  
Author(s):  
Andrei Agrachev ◽  
Paul W. Y. Lee
Keyword(s):  
Riemannian Manifolds ◽  
Three Dimensional ◽  
Comparison Theorems
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