On local isometric immersions into complex and quaternionic projective spaces
Archivum Mathematicum (2011)
- Volume: 047, Issue: 4, page 251-256
- ISSN: 0044-8753
Access Full Article
top
Abstract
topHow to cite
topRivertz, Hans Jakob. "On local isometric immersions into complex and quaternionic projective spaces." Archivum Mathematicum 047.4 (2011): 251-256. <http://eudml.org/doc/246180>.
@article{Rivertz2011,
abstract = {We will prove that if an open subset of $\mathbb \{C\}\{\}P^\{n\}$ is isometrically immersed into $\mathbb \{C\}\{\}P^\{m\}$, with $m<(4/3)n-2/3$, then the image is totally geodesic. We will also prove that if an open subset of $\mathbb \{H\}\{\}P^\{n\}$ isometrically immersed into $\mathbb \{H\}\{\}P^\{m\}$, with $m<(4/3)n-5/6$, then the image is totally geodesic.},
author = {Rivertz, Hans Jakob},
journal = {Archivum Mathematicum},
keywords = {submanifolds; homogeneous spaces; symmetric spaces; submanifold; homogeneous space; symmetric space},
language = {eng},
number = {4},
pages = {251-256},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On local isometric immersions into complex and quaternionic projective spaces},
url = {http://eudml.org/doc/246180},
volume = {047},
year = {2011},
}
TY - JOUR
AU - Rivertz, Hans Jakob
TI - On local isometric immersions into complex and quaternionic projective spaces
JO - Archivum Mathematicum
PY - 2011
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 047
IS - 4
SP - 251
EP - 256
AB - We will prove that if an open subset of $\mathbb {C}{}P^{n}$ is isometrically immersed into $\mathbb {C}{}P^{m}$, with $m<(4/3)n-2/3$, then the image is totally geodesic. We will also prove that if an open subset of $\mathbb {H}{}P^{n}$ isometrically immersed into $\mathbb {H}{}P^{m}$, with $m<(4/3)n-5/6$, then the image is totally geodesic.
LA - eng
KW - submanifolds; homogeneous spaces; symmetric spaces; submanifold; homogeneous space; symmetric space
UR - http://eudml.org/doc/246180
ER -
References
top- Agaoka, Y., A note on local isometric imbeddings of complex projective spaces, J. Math. Kyoto Univ. 27 (3) (1987), 501–505. (1987) Zbl0633.53080MR0910231
- Agaoka, Y., Kaneda, E., 10.2748/tmj/1178228907, Tôhoku Math. J. 36 (1984), 107–140. (1984) Zbl0533.53052MR0733623DOI10.2748/tmj/1178228907
- Bourguignon, J., Karcher, H., Curvature operators pinching estimates and geometric examples, Ann. Sci. École Norm. Sup. (4) 11 (1978), 71–92. (1978) Zbl0386.53031MR0493867
- Dajczer, M., Rodriguez, L., On isometric immersions into complex space forms, VIII School on Differential Geometry (Portuguese) (Campinas, 1992), vol. 4, Mat. Contemp., 1993, pp. 95–98. (1993) Zbl0852.53044MR1302494
- Dajczer, M., Rodriguez, L., 10.1007/BF01459781, Math. Ann. 299 (1994), 223–230. (1994) Zbl0806.53019MR1275765DOI10.1007/BF01459781
- Gray, A., A note on manifolds whose holonomy group is a subgroup of , Michigan Math. J. 16 (1969), 125–128. (1969) MR0244913
- Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, San Francisco and London, 1978, Ch. 4. (1978) Zbl0451.53038MR0514561
- Küpelî, D. N., Notes on totally geodesic Hermitian subspaces of indefinite Kähler manifolds, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 43 (1) (1995), 1–7. (1995) MR1338255
- Rivertz, H. J., On isometric and conformal immersions into Riemannian spaces, Ph.D. thesis, Department of Mathematics, University of Oslo, 1999. (1999)
- Tomter, P., Isometric immersions into complex projective space, Lie groups, geometric structures and differential equations—one hundred years after Sophus Lie, vol. 37, Adv. Stud. Pure Math., 2002, pp. 367–396. (2002) Zbl1043.53047MR1980909
- Wolf, J. A., 10.1007/BF02392195, Acta Math. 152 (1984), 141–152. (1984) Zbl0539.53037MR0736216DOI10.1007/BF02392195
NotesEmbed ?
topYou 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.