211 research outputs found
On Rigidity of Generalized Conformal Structures
The classical Liouville Theorem on conformal transformations determines local
conformal transformations on the Euclidean space of dimension . Its
natural adaptation to the general framework of Riemannian structures is the
2-rigidity of conformal transformations, that is such a transformation is fully
determined by its 2-jet at any point. We prove here a similar rigidity for
generalized conformal structures defined by giving a one parameter family of
metrics (instead of scalar multiples of a given one) on each tangent space
Leafwise Holomorphic Functions
It is a well-known and elementary fact that a holomorphic function on a
compact complex manifold without boundary is necessarily constant. The purpose
of the present article is to investigate whether, or to what extent, a similar
property holds in the setting of holomorphically foliated spaces
Real and discrete holomorphy : Introduction to an algebraic approach
We consider spaces for which there is a notion of harmonicity for complex
valued functions defined on them. For instance, this is the case of Riemannian
manifolds on one hand, and (metric) graphs on the other hand. We observe that
it is then possible to define an "amazing" notion of holomorphic functions on
them, and show how rigid it is in some cases
Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes, Application to the Minkowski problem in the Minkowski space
We study the existence of surfaces with constant or prescribed Gauss
curvature in certain Lorentzian spacetimes. We prove in particular that every
(non-elementary) 3-dimensional maximal globally hyperbolic spatially compact
spacetime with constant non-negative curvature is foliated by compact spacelike
surfaces with constant Gauss curvature. In the constant negative curvature
case, such a foliation exists outside the convex core. The existence of these
foliations, together with a theorem of C. Gerhardt, yield several corollaries.
For example, they allow to solve the Minkowski problem in the 3-dimensional
Minkowski space for datas that are invariant under the action of a co-compact
Fuchsian group
On Lorentz dynamics : From group actions to warped products via homogeneous spaces
We show a geometric rigidity of isometric actions of non compact (semisimple)
Lie groups on Lorentz manifolds. Namely, we show that the manifold has a warped
product structure of a Lorentz manifold with constant curvature by a Riemannian
manifold
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