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Calibrated geodesic foliations of hyperbolic space


Authors: Yamile Godoy and Marcos Salvai
Journal: Proc. Amer. Math. Soc. 144 (2016), 359-367
MSC (2010): Primary 53C38, 53C12, 53C22, 53C50
DOI: https://doi.org/10.1090/proc/12834
Published electronically: July 30, 2015
MathSciNet review: 3415602
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Abstract: Let $ H$ be the hyperbolic space of dimension $ n+1$. A geodesic foliation of $ H$ is given by a smooth unit vector field on $ H$ all of whose integral curves are geodesics. Each geodesic foliation of $ H$ determines an $ n$-dimensional submanifold of the $ 2n$-dimensional manifold $ \mathcal {L}$ of all the oriented geodesics of $ H$ (up to orientation preserving reparametrizations). The space $ \mathcal {L}$ has a canonical split semi-Riemannian metric induced by the Killing form of the isometry group of $ H$. Using a split special Lagrangian calibration, we study the volume maximization problem for a certain class of geometrically distinguished geodesic foliations, whose corresponding submanifolds of $ \mathcal {L}$ are space-like.


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Additional Information

Yamile Godoy
Affiliation: FaMAF-CIEM, Ciudad Universitaria, 5000 Córdoba, Argentina
Email: ygodoy@famaf.unc.edu.ar

Marcos Salvai
Affiliation: FaMAF-CIEM, Ciudad Universitaria, 5000 Córdoba, Argentina
Email: salvai@famaf.unc.edu.ar

DOI: https://doi.org/10.1090/proc/12834
Keywords: Split special Lagrangian calibration, geodesic foliation, hyperbolic space, space of geodesics
Received by editor(s): November 28, 2014
Published electronically: July 30, 2015
Additional Notes: The authors were partially supported by CONICET, FONCyT, SECyT (UNC)
Communicated by: Michael Wolf
Article copyright: © Copyright 2015 American Mathematical Society