Differential geometry of grassmannians and the Plücker map

• Autorzy: Sasha Anan’in , Carlos Grossi
• Języki publikacji: EN

Abstrakty

EN

Using the Plücker map between grassmannians, we study basic aspects of classic grassmannian geometries. For ‘hyperbolic’ grassmannian geometries, we prove some facts (for instance, that the Plücker map is a minimal isometric embedding) that were previously known in the ‘elliptic’ case.

Słowa kluczowe

EN

Classic geometries; Plücker map; Hyperbolic geometry

Kategorie tematyczne

• 51M10: Hyperbolic and elliptic geometries (general) and generalizations
• 53A35: Non-Euclidean differential geometry
• 53A20: Projective differential geometry

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 3
• Strony: 873-884

Daty

wydano

2012-06-01

online

2012-03-24

Twórcy

autor

Sasha Anan’in

• Universidade Estadual de Campinas

autor

Carlos Grossi

• Universidade de São Paulo

Bibliografia

• [1] Anan’in S., Gonçalves E.C.B., Grossi C.H., Grassmannians and conformal structure on absolutes, preprint available at http://arxiv.org/abs/0907.4469
• [2] Anan’in S., Grossi C., Coordinate-free classic geometries, Mosc. Math. J., 2011, 11(4), 633–655
• [3] Borisenko A.A., Nikolaevskiı Yu.A., Grassmann manifolds and Grassmann image of submanifolds, Russian Math. Surveys, 1991, 46(2), 45–94 http://dx.doi.org/10.1070/RM1991v046n02ABEH002742
• [4] do Carmo M.P., Riemannian Geometry, Math. Theory Appl., Birkhäuser, Boston, 1992
• [5] Gromov M., Lawson H.B. Jr., Thurston W., Hyperbolic 4-manifolds and conformally flat 3-manifolds, Inst. Hautes Études Sci. Publ. Math., 1988, 68, 27–45 http://dx.doi.org/10.1007/BF02698540
• [6] Guilfoyle B., Klingenberg W., Proof of the Carathéodory conjecture by mean curvature flow in the space of oriented affine lines, preprint available at http://arxiv.org/abs/0808.0851
• [7] Kobayashi S., Nomizu K., Foundations of Differential Geometry. II, Interscience Tracts in Pure and Applied Mathematics, 15(2), John Wiley & Sons, New York-London-Sydney, 1969
• [8] Kuiper N.H., Hyperbolic 4-manifolds and tesselations, Inst. Hautes Études Sci. Publ. Math., 1988, 68, 47–76 http://dx.doi.org/10.1007/BF02698541
• [9] Luo F., Constructing conformally flat structures on some Seifert fibred 3-manifolds, Math. Ann., 1992, 294(3), 449–458 http://dx.doi.org/10.1007/BF01934334

Identyfikatory

DOI

10.2478/s11533-012-0021-y