Euler characters and submanifolds of constant positive curvature

Author:
John Douglas Moore

Journal:
Trans. Amer. Math. Soc. **354** (2002), 3815-3834

MSC (2000):
Primary 53C40; Secondary 57R20

Published electronically:
May 7, 2002

MathSciNet review:
1911523

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This article develops methods for studying the topology of submanifolds of constant positive curvature in Euclidean space. It proves that if is an -dimensional compact connected Riemannian submanifold of constant positive curvature in , then must be simply connected. It also gives a conformal version of this theorem.

**1.**Ichiro Amemiya and Kazuo Masuda,*On Joris’ theorem on differentiability of functions*, Kodai Math. J.**12**(1989), no. 1, 92–97. MR**987144**, 10.2996/kmj/1138038992**2.**Jan Boman,*Differentiability of a function and of its compositions with functions of one variable*, Math. Scand.**20**(1967), 249–268. MR**0237728****3.**Jeff Cheeger and James Simons,*Differential characters and geometric invariants*, Geometry and topology (College Park, Md., 1983/84) Lecture Notes in Math., vol. 1167, Springer, Berlin, 1985, pp. 50–80. MR**827262**, 10.1007/BFb0075216**4.**Shiing-shen Chern,*A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds*, Ann. of Math. (2)**45**(1944), 747–752. MR**0011027****5.**Shiing-shen Chern and Nicolaas H. Kuiper,*Some theorems on the isometric imbedding of compact Riemann manifolds in euclidean space*, Ann. of Math. (2)**56**(1952), 422–430. MR**0050962****6.**Shiing Shen Chern and James Simons,*Characteristic forms and geometric invariants*, Ann. of Math. (2)**99**(1974), 48–69. MR**0353327****7.**Herbert Federer,*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325****8.**Shoshichi Kobayashi and Katsumi Nomizu,*Foundations of differential geometry. Vol I*, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR**0152974**

Shoshichi Kobayashi and Katsumi Nomizu,*Foundations of differential geometry. Vol. II*, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR**0238225****9.**John J. Millson,*Examples of nonvanishing Chern-Simons invariants*, J. Differential Geometry**10**(1975), no. 4, 589–600. MR**0394695****10.**John Douglas Moore,*Conformally flat submanifolds of Euclidean space*, Math. Ann.**225**(1977), no. 1, 89–97. MR**0431046****11.**John Douglas Moore,*Submanifolds of constant positive curvature. I*, Duke Math. J.**44**(1977), no. 2, 449–484. MR**0438256****12.**John Douglas Moore,*Codimension two submanifolds of positive curvature*, Proc. Amer. Math. Soc.**70**(1978), no. 1, 72–74. MR**487560**, 10.1090/S0002-9939-1978-0487560-8**13.**John Douglas Moore,*On conformal immersions of space forms*, Global differential geometry and global analysis (Berlin, 1979) Lecture Notes in Math., vol. 838, Springer, Berlin, 1981, pp. 203–210. MR**636283****14.**John Douglas Moore,*On extendability of isometric immersions of spheres*, Duke Math. J.**85**(1996), no. 3, 685–699. MR**1422362**, 10.1215/S0012-7094-96-08526-9**15.**M. S. Narasimhan and S. Ramanan,*Existence of universal connections*, Amer. J. Math.**83**(1961), 563–572. MR**0133772****16.**Barrett O’Neill,*Umbilics of constant curvature immersions*, Duke Math. J.**32**(1965), 149–159. MR**0180951**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
53C40,
57R20

Retrieve articles in all journals with MSC (2000): 53C40, 57R20

Additional Information

**John Douglas Moore**

Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 93106

Email:
moore@math.ucsb.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-03043-X

Received by editor(s):
March 28, 2001

Published electronically:
May 7, 2002

Article copyright:
© Copyright 2002
American Mathematical Society