Regions Where the Exponential Map at Regular Points of Sub-Riemannian Manifolds is a Local Diffeomorphism

2009 ◽  
Vol 15 (1) ◽  
pp. 133-156 ◽  
Author(s):  
Marcos M. Diniz ◽  
José M. M. Veloso
Keyword(s):  
Riemannian Manifolds ◽  
Exponential Map ◽  
Local Diffeomorphism

scholarly journals On the Singularities of the Exponential Map in Infinite Dimensional Riemannian Manifolds

2006 ◽  
Vol 336 (2) ◽  
pp. 247-267 ◽  
Author(s):  
Leonardo Biliotti ◽  
Ruy Exel ◽  
Paolo Piccione ◽  
Daniel V. Tausk
Keyword(s):  
Riemannian Manifolds ◽  
Exponential Map ◽  
Infinite Dimensional

scholarly journals L2 Extension for jets of holomorphic sections of a Hermitian line Bundle

2005 ◽  
Vol 180 ◽  
pp. 1-34 ◽  
Author(s):  
Dan Popovici
Keyword(s):  
Line Bundle ◽  
Riemannian Manifolds ◽  
Technical Difficulty ◽  
Holomorphic Section ◽  
Exponential Map ◽  
Extension Theorems ◽  
Holomorphic Sections ◽  
Final Estimate ◽  
Arbitrary Compact Subset ◽  
Complete Riemannian Manifolds

AbstractLet (X, ω) be a weakly pseudoconvex Kähler manifold, Y ⊂ X a closed submanifold defined by some holomorphic section of a vector bundle over X, and L a Hermitian line bundle satisfying certain positivity conditions. We prove that for any integer k > 0, any section of the jet sheaf which satisfies a certain L2 condition, can be extended into a global holomorphic section of L over X whose L2 growth on an arbitrary compact subset of X is under control. In particular, if Y is merely a point, this gives the existence of a global holomorphic function with an L2 norm under control and with prescribed values for all its derivatives up to order k at that point. This result generalizes the L2 extension theorems of Ohsawa-Takegoshi and of Manivel to the case of jets of sections of a line bundle. A technical difficulty is to achieve uniformity in the constant appearing in the final estimate. To this end, we make use of the exponential map and of a Rauch-type comparison theorem for complete Riemannian manifolds.

Convex Hamiltonians without conjugate points

1999 ◽  
Vol 19 (4) ◽  
pp. 901-952 ◽  
Author(s):  
GONZALO CONTRERAS ◽  
RENATO ITURRIAGA
Keyword(s):  
Periodic Orbit ◽  
Energy Level ◽  
Conjugate Points ◽  
Exponential Map ◽  
Metric Entropy ◽  
Hamiltonian Flow ◽  
Local Diffeomorphism ◽  
Index Form ◽  
Exponential Maps ◽  
Contact Flows

We construct the Green bundles for an energy level without conjugate points of a convex Hamiltonian. In this case we give a formula for the metric entropy of the Liouville measure and prove that the exponential map is a local diffeomorphism. We prove that the Hamiltonian flow is Anosov if and only if the Green bundles are transversal. Using the Clebsch transformation of the index form we prove that if the unique minimizing measure of a generic Lagrangian is supported on a periodic orbit, then it is a hyperbolic periodic orbit.We also show some examples of differences with the behaviour of a geodesic flow without conjugate points, namely: (non-contact) flows and periodic orbits without invariant transversal bundles, segments without conjugate points but with crossing solutions and non-surjective exponential maps.

On Ricci Riemannian Manifolds

2010 ◽  
Vol 0 (-1) ◽  
pp. 437-446 ◽  
Author(s):  
S. K. Saha
Keyword(s):  
Riemannian Manifolds

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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