|
Homology vanishing theorems for submanifolds
Author(s):
Theodoros
Vlachos
Journal:
Proc. Amer. Math. Soc.
135
(2007),
2607-2617.
MSC (2000):
Primary 53C40;
Secondary 53C20.
Posted:
March 30, 2007
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
We relate intrinsic and extrinsic curvature invariants to the homology groups of submanifolds in space forms of nonnegative curvature. More precisely, we provide bounds for the squared length of the second fundamental form, or the Ricci curvature in terms of the mean curvature, which force homology to vanish in a range of intermediate dimensions. Moreover, we give examples which show that these conditions are sharp.
References:
-
- 1.
- K.D. Elworthy and S. Rosenberg, Homotopy and homology vanishing theorems and the stability of stochastic flows, Geom. Funct. Anal. 6 (1996), 51-78. MR 1371231 (97a:58198)
- 2.
- H. Federer and W.H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458-520. MR 0123260 (23:A588)
- 3.
- Th. Hasanis and Th. Vlachos, Ricci curvature and minimal submanifolds, Pacific J. Math. 197 (2001), 13-24. MR 1810205 (2001j:53077)
- 4.
- H.B. Lawson and J. Simons, On stable currents and their application to global problems in real and complex geometry, Ann. of Math. (2) 98 (1973), 427-450. MR 0324529 (48:2881)
- 5.
- K. Shiohama and H. Xu, The topological sphere theorem for complete submanifolds, Compositio Math. 107 (1997), 221-232. MR 1458750 (98i:53080)
- 6.
- D. Sjerve, Homology spheres which are covered by spheres , J. London Math. Soc. (2) 6 (1973), 333-336. MR 0310895 (46:9993)
- 7.
- Th. Vlachos, A sphere theorem for odd-dimensional submanifolds of spheres, Proc. Amer. Math. Soc. 130 (2002), 167-173. MR 1855635 (2003c:53083)
- 8.
- N.R. Wallach, Minimal immersions of symmetric spaces into spheres, ``Symmetric spaces", Ed. Boothby and Weiss, Dekker, New York, 1972, pp. 1-40. MR 0407774 (53:11545)
- 9.
- Y.L. Xin, An application of integral currents to the vanishing theorems, Sci. Sinica Ser. A 27 (1984), 233-241. MR 763966 (86b:49060)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
53C40,
53C20.
Retrieve articles in all Journals with MSC
(2000):
53C40,
53C20.
Additional Information:
Theodoros
Vlachos
Affiliation:
Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
Email:
tvlachos@uoi.gr
DOI:
10.1090/S0002-9939-07-08901-0
PII:
S 0002-9939(07)08901-0
Keywords:
Ricci curvature,
length of the second fundamental form,
mean curvature,
homology groups
Received by editor(s):
May 9, 2006
Posted:
March 30, 2007
Communicated by:
Jon G. Wolfson
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org