Rigidity of spheres in Riemannian manifolds and a non-embedding theorem

2001 ◽  
Vol 32 (2) ◽  
pp. 159-171 ◽  
Author(s):  
Alireza Ranjbar-Motlagh
Keyword(s):  
Riemannian Manifolds ◽  
Embedding Theorem

scholarly journals A tour through δ-invariants: From Nash’s embedding theorem to ideal immersions, best ways of living and beyond

2013 ◽  
Vol 94 (108) ◽  
pp. 67-80 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Riemannian Manifolds ◽  
Lagrangian Submanifolds ◽  
Embedding Theorem ◽  
The Many ◽  
Optimal Inequalities

First I will explain my motivation to introduce the ?-invariants for Riemannian manifolds. I will also recall the notions of ideal immersions and best ways of living. Then I will present a few of the many applications of ?-invariants in several areas in mathematics. Finally, I will present two optimal inequalities involving ?-invariants for Lagrangian submanifolds obtained very recently in joint papers with F. Dillen, J. Van der Veken and L. Vrancken.

scholarly journals On Weyl’s Embedding Problem in Riemannian Manifolds

2018 ◽  
Vol 2020 (11) ◽  
pp. 3229-3259 ◽  
Author(s):  
Siyuan Lu
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Space Form ◽  
A Priori Estimates ◽  
A Priori ◽  
Three Dimensional ◽  
Embedding Theorem ◽  
Embedding Problem ◽  
Priori Estimates ◽  
Geometric Assumption

Abstract We consider a priori estimates of Weyl’s embedding problem of $(\mathbb{S}^2, g)$ in general three-dimensional Riemannian manifold $(N^3, \bar g)$. We establish interior $C^2$ estimate under natural geometric assumption. Together with a recent work by Li and Wang [18], we obtain an isometric embedding of $(\mathbb{S}^2,g)$ in Riemannian manifold. In addition, we reprove Weyl’s embedding theorem in space form under the condition that $g\in C^2$ with $D^2g$ Dini continuous.

Some Newman-type theorems for maps from Riemannian manifolds into manifolds

1989 ◽  
Vol 111 (1-2) ◽  
pp. 53-59
Author(s):  
Duan Hai-bao
Keyword(s):  
Lower Bound ◽  
Riemannian Manifolds ◽  
Embedding Theorem ◽  
Major Purpose ◽  
Image Position

SynopsisLetf: Mm→Nnbe a map from a Riemannianm-manifold(Mm, d)into ann-manifold Nn. The major purpose of this paper is to give a lower bound for the numberby examining the behaviour of the cohomology homomorphisms induced byf. This idea will be used to generalise the classical Newman theorem and present a geometric background for a well-known non-embedding theorem in topology.

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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