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Exotic Deformations of Calabi-Yau Manifolds

Paolo de Bartolomeis, Adriano Tomassini (2013)

Annales de l’institut Fourier

We introduce Quantum Inner State manifolds (QIS manifolds) as (compact) 2 n -dimensional symplectic manifolds ( M , κ ) endowed with a κ -tamed almost complex structure J and with a nowhere vanishing and normalized section ϵ of the bundle Λ J n , 0 ( M ) satisfying the condition ¯ J ϵ = 0 .We study the moduli space 𝔐 of QIS deformations of a given Calabi-Yau manifold, computing its tangent space and showing that 𝔐 is non obstructed. Finally, we present several examples of QIS manifolds.

Explicit geodesic graphs on some H-type groups

Dušek, Zdeněk (2002)

Proceedings of the 21st Winter School "Geometry and Physics"

A homogeneous Riemannian manifold M = G / H is called a “g.o. space” if every geodesic on M arises as an orbit of a one-parameter subgroup of G . Let M = G / H be such a “g.o. space”, and m an Ad ( H ) -invariant vector subspace of Lie ( G ) such that Lie ( G ) = m Lie ( H ) . A geodesic graph is a map ξ : m Lie ( H ) such that t exp ( t ( X + ξ ( X ) ) ) ( e H ) is a geodesic for every X m { 0 } . The author calculates explicitly such geodesic graphs for certain special 2-step nilpotent Lie groups. More precisely, he deals with “generalized Heisenberg groups” (also known as “H-type groups”) whose center has...

Exploration d’un mode d’écriture de la généralité : l’article de Poincaré sur les lignes géodésiques des surfaces convexes (1905)

Anne Robadey (2004)

Revue d'histoire des mathématiques

L’analyse de l’article de Poincaré sur les géodésiques fait apparaître qu’il entretient des liens complexes avec les travaux antérieurs de Poincaré en mécanique céleste. Nous montrerons que le problème des géodésiques des surfaces convexes est traité comme un paradigme grâce auquel Poincaré explicite une méthode qui n’était présentée qu’à l’état d’ébauche dans ses ouvrages de mécanique céleste. Cette étude de cas permet ainsi de mettre en évidence l’utilisation par Poincaré d’une technique d’écriture...

Extended Derdziński-Shen theorem for curvature tensors

Carlo Alberto Mantica, Luca Guido Molinari (2012)

Colloquium Mathematicae

We extend a remarkable theorem of Derdziński and Shen, on the restrictions imposed on the Riemann tensor by the existence of a nontrivial Codazzi tensor. We show that the Codazzi equation can be replaced by a more general algebraic condition. The resulting extension applies both to the Riemann tensor and to generalized curvature tensors.

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