## Calibrated geodesic foliations of hyperbolic space

HTML articles powered by AMS MathViewer

- by Yamile Godoy and Marcos Salvai PDF
- Proc. Amer. Math. Soc.
**144**(2016), 359-367 Request permission

## 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.## References

- M. Berger,
*Quelques problèmes de géométrie riemannienne ou deux variations sur les espaces symétriques compacts de rang un*, Enseign. Math. (2)**16**(1970), 73–96 (French). MR**262978** - Jiri Dadok and Reese Harvey,
*Calibrations on $\textbf {R}^{6}$*, Duke Math. J.**50**(1983), no. 4, 1231–1243. MR**726326**, DOI 10.1215/S0012-7094-83-05053-6 - Nikos Georgiou and Brendan Guilfoyle,
*On the space of oriented geodesics of hyperbolic 3-space*, Rocky Mountain J. Math.**40**(2010), no. 4, 1183–1219. MR**2718810**, DOI 10.1216/RMJ-2010-40-4-1183 - Herman Gluck, Dana Mackenzie, and Frank Morgan,
*Volume-minimizing cycles in Grassmann manifolds*, Duke Math. J.**79**(1995), no. 2, 335–404. MR**1344765**, DOI 10.1215/S0012-7094-95-07909-5 - Herman Gluck, Frank Morgan, and Wolfgang Ziller,
*Calibrated geometries in Grassmann manifolds*, Comment. Math. Helv.**64**(1989), no. 2, 256–268. MR**997365**, DOI 10.1007/BF02564674 - Herman Gluck and Frank W. Warner,
*Great circle fibrations of the three-sphere*, Duke Math. J.**50**(1983), no. 1, 107–132. MR**700132** - Herman Gluck and Wolfgang Ziller,
*On the volume of a unit vector field on the three-sphere*, Comment. Math. Helv.**61**(1986), no. 2, 177–192. MR**856085**, DOI 10.1007/BF02621910 - Y. Godoy and M. Salvai,
*Global smooth geodesic foliations of the hyperbolic space*, To appear in Math. Z. DOI: 10.1007/s00209-015-1474-z. - Reese Harvey and H. Blaine Lawson Jr.,
*Calibrated geometries*, Acta Math.**148**(1982), 47–157. MR**666108**, DOI 10.1007/BF02392726 - Reese Harvey and H. Blaine Lawson Jr.,
*Split special Lagrangian geometry*, Metric and Differential Geometry: The Jeff Cheeger Anniversary Volume (X. Dai and X. Rong, eds.), Springer, 2012, pp. 43–89. - Young-Heon Kim, Robert J. McCann, and Micah Warren,
*Pseudo-Riemannian geometry calibrates optimal transportation*, Math. Res. Lett.**17**(2010), no. 6, 1183–1197. MR**2729641**, DOI 10.4310/MRL.2010.v17.n6.a16 - Jack Mealy,
*Volume maximization in semi-Riemannian manifolds*, Indiana Univ. Math. J.**40**(1991), no. 3, 793–814. MR**1129330**, DOI 10.1512/iumj.1991.40.40036 - Frank Morgan,
*On the singular structure of three-dimensional, area-minimizing surfaces*, Trans. Amer. Math. Soc.**276**(1983), no. 1, 137–143. MR**684498**, DOI 10.1090/S0002-9947-1983-0684498-4 - Frank Morgan,
*Geometric measure theory*, 4th ed., Elsevier/Academic Press, Amsterdam, 2009. A beginner’s guide. MR**2455580** - Marcos Salvai,
*A two point calibration on an Sp(1) bundle over the three-sphere*, J. Differential Geom.**59**(2001), no. 3, 523–533. MR**1916954** - Marcos Salvai,
*Some geometric characterizations of the Hopf fibrations of the three-sphere*, Monatsh. Math.**147**(2006), no. 2, 173–177. MR**2216560**, DOI 10.1007/s00605-005-0327-y - Marcos Salvai,
*On the geometry of the space of oriented lines of the hyperbolic space*, Glasg. Math. J.**49**(2007), no. 2, 357–366. MR**2347266**, DOI 10.1017/S0017089507003710 - Micah Warren,
*Calibrations associated to Monge-Ampère equations*, Trans. Amer. Math. Soc.**362**(2010), no. 8, 3947–3962. MR**2608392**, DOI 10.1090/S0002-9947-10-05109-3

## Additional Information

**Yamile Godoy**- Affiliation: FaMAF - CIEM, Ciudad Universitaria, 5000 Córdoba, Argentina
- MR Author ID: 1043601
- Email: ygodoy@famaf.unc.edu.ar
**Marcos Salvai**- Affiliation: FaMAF - CIEM, Ciudad Universitaria, 5000 Córdoba, Argentina
- MR Author ID: 603972
- Email: salvai@famaf.unc.edu.ar
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 359-367 - MSC (2010): Primary 53C38, 53C12, 53C22, 53C50
- DOI: https://doi.org/10.1090/proc/12834
- MathSciNet review: 3415602