scholarly journals Smooth approximation of Lipschitz functions on Riemannian manifolds

2007 ◽  
Vol 326 (2) ◽  
pp. 1370-1378 ◽  
Author(s):  
D. Azagra ◽  
J. Ferrera ◽  
F. López-Mesas ◽  
Y. Rangel
Keyword(s):  
Riemannian Manifolds ◽  
Lipschitz Functions ◽  
Smooth Approximation

scholarly journals Smooth Approximation of Lipschitz Functions on Finsler Manifolds

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
M. I. Garrido ◽  
J. A. Jaramillo ◽  
Y. C. Rangel
Keyword(s):  
Lipschitz Function ◽  
Composition Operators ◽  
Lipschitz Functions ◽  
Finsler Manifold ◽  
Positive Continuous Function ◽  
Smooth Approximation ◽  
Finsler Manifolds ◽  
Lipschitz Constants ◽  
Bounded Derivative

We study the smooth approximation of Lipschitz functions on Finsler manifolds, keeping control on the corresponding Lipschitz constants. We prove that, given a Lipschitz functionf:M→ℝdefined on a connected, second countable Finsler manifoldM, for each positive continuous functionε:M→(0,∞)and eachr>0, there exists aC1-smooth Lipschitz functiong:M→ℝsuch that|f(x)-g(x)|≤ε(x), for everyx∈M, andLip(g)≤Lip(f)+r. As a consequence, we derive a completeness criterium in the class of what we call quasi-reversible Finsler manifolds. Finally, considering the normed algebraCb1(M)of allC1functions with bounded derivative on a complete quasi-reversible Finsler manifoldM, we obtain a characterization of algebra isomorphismsT:Cb1(N)→Cb1(M)as composition operators. From this we obtain a variant of Myers-Nakai Theorem in the context of complete reversible Finsler manifolds.

Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group

2013 ◽  
Vol 49 (3-4) ◽  
pp. 1279-1308 ◽  
Author(s):  
Giovanna Citti ◽  
Maria Manfredini ◽  
Andrea Pinamonti ◽  
Francesco Serra Cassano
Keyword(s):  
Heisenberg Group ◽  
Lipschitz Functions ◽  
Smooth Approximation

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
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