Existence of compact flat Riemannian manifolds with the first Betti number equal to zero

1986 ◽  
pp. 391-393
Author(s):  
Andrzej Szczepański
Keyword(s):  
Riemannian Manifolds ◽  
Betti Number

scholarly journals On the structure of Riemannian manifolds of almost nonnegative Ricci curvature

2004 ◽  
Vol 2004 (39) ◽  
pp. 2085-2090
Author(s):  
Gabjin Yun
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Betti Number ◽  
Compact Riemannian Manifold ◽  
Nonnegative Real Number ◽  
Sphere Bundle ◽  
Finite Cover ◽  
Nonnegative Ricci Curvature ◽  
Bounded Curvature

We study the structure of manifolds with almost nonnegative Ricci curvature. We prove a compact Riemannian manifold with bounded curvature, diameter bounded from above, and Ricci curvature bounded from below by an almost nonnegative real number such that the first Betti number havingcodimension two is an infranilmanifold or a finite cover is a sphere bundle over a torus. Furthermore, if we assume the Ricci curvature is bounded and volume is bounded from below, then the manifold must be an infranilmanifold.

scholarly journals On Compact Riemannian Manifolds with Zero Ricci Curvature

1956 ◽  
Vol 10 (3) ◽  
pp. 131-133 ◽  
Author(s):  
T. J. Willmore
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Betti Number ◽  
Positive Definite ◽  
Arbitrary Signature ◽  
Local Result

In this note I prove the following result:—Theorem. A compact, orientable, Riemannian manifold Mn, with positive definite metric and zero Ricci curvature, is flat if the first Betti number R1 exceeds n — 4.In this statement of the theorem it is assumed that the dimensions of Mn are not less than four. If this is not the case, the result is still valid but appears as a purely local result and is true for a metric of arbitrary signature.

On the equicontinuity of families of inverse mappings of Riemannian manifolds

2019 ◽  
Vol 16 (4) ◽  
pp. 557-566
Author(s):  
Denis Ilyutko ◽  
Evgenii Sevost'yanov
Keyword(s):  
Riemannian Manifolds ◽  
Type Inequality ◽  
The Given ◽  
Range Of Values

We study homeomorphisms of Riemannian manifolds with unbounded characteristic such that the inverse mappings satisfy the Poletsky-type inequality. It is established that their families are equicontinuous if the function Q which is related to the Poletsky inequality and is responsible for a distortion of the modulus, is integrable in the given domain, here the original manifold is connected and the domain of definition and the range of values of mappings have compact closures.

Ball-homogeneous and disk-homogeneous Riemannian manifolds

1982 ◽  
Vol 180 (4) ◽  
pp. 429-444 ◽  
Author(s):  
Old?ich Kowalski ◽  
Lieven Vanhecke
Keyword(s):  
Riemannian Manifolds ◽  
Homogeneous Riemannian Manifolds

scholarly journals A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

2021 ◽  
Author(s):  
Alessandro Goffi ◽  
Francesco Pediconi
Keyword(s):  
Maximum Principle ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Strong Maximum Principle ◽  
Second Order ◽  
Fully Nonlinear Equations ◽  
Nonlinear Operators ◽  
Fully Nonlinear ◽  
Second Order Equations ◽  
Curvature Type

AbstractWe investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and $$\infty $$ ∞ -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

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