Differential geometry of grassmannians and the Plücker map

Sasha Anan’in; Carlos Grossi

Open Mathematics (2012)

  • Volume: 10, Issue: 3, page 873-884
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

Sasha Anan’in, and Carlos Grossi. "Differential geometry of grassmannians and the Plücker map." Open Mathematics 10.3 (2012): 873-884. <http://eudml.org/doc/268939>.

@article{SashaAnan2012,
abstract = {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.},
author = {Sasha Anan’in, Carlos Grossi},
journal = {Open Mathematics},
keywords = {Classic geometries; Plücker map; Hyperbolic geometry; classic geometries; hyperbolic geometry},
language = {eng},
number = {3},
pages = {873-884},
title = {Differential geometry of grassmannians and the Plücker map},
url = {http://eudml.org/doc/268939},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Sasha Anan’in
AU - Carlos Grossi
TI - Differential geometry of grassmannians and the Plücker map
JO - Open Mathematics
PY - 2012
VL - 10
IS - 3
SP - 873
EP - 884
AB - 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.
LA - eng
KW - Classic geometries; Plücker map; Hyperbolic geometry; classic geometries; hyperbolic geometry
UR - http://eudml.org/doc/268939
ER -

References

top
  1. [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. [2] Anan’in S., Grossi C., Coordinate-free classic geometries, Mosc. Math. J., 2011, 11(4), 633–655 Zbl1256.53014
  3. [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 Zbl0742.53017
  4. [4] do Carmo M.P., Riemannian Geometry, Math. Theory Appl., Birkhäuser, Boston, 1992 
  5. [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 Zbl0692.57012
  6. [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 Zbl1060.51015
  7. [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 Zbl0175.48504
  8. [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 Zbl0692.57013
  9. [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 Zbl0758.57009

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.