Euler characters and submanifolds of constant positive curvature
Author:
John Douglas Moore
Journal:
Trans. Amer. Math. Soc. 354 (2002), 38153834
MSC (2000):
Primary 53C40; Secondary 57R20
Published electronically:
May 7, 2002
MathSciNet review:
1911523
Fulltext 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 (90a:26005), http://dx.doi.org/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
(38 #6009)
 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
(87g:53059), http://dx.doi.org/10.1007/BFb0075216
 4.
Shiingshen
Chern, A simple intrinsic proof of the GaussBonnet formula for
closed Riemannian manifolds, Ann. of Math. (2) 45
(1944), 747–752. MR 0011027
(6,106a)
 5.
Shiingshen
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
(14,408e)
 6.
Shiing
Shen Chern and James
Simons, Characteristic forms and geometric invariants, Ann. of
Math. (2) 99 (1974), 48–69. MR 0353327
(50 #5811)
 7.
Herbert
Federer, Geometric measure theory, Die Grundlehren der
mathematischen Wissenschaften, Band 153, SpringerVerlag New York Inc., New
York, 1969. MR
0257325 (41 #1976)
 8.
Shoshichi
Kobayashi and Katsumi
Nomizu, Foundations of differential geometry. Vol I,
Interscience Publishers, a division of John Wiley & Sons, New
YorkLondon, 1963. MR 0152974
(27 #2945)
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
YorkLondonSydney, 1969. MR 0238225
(38 #6501)
 9.
John
J. Millson, Examples of nonvanishing ChernSimons invariants,
J. Differential Geometry 10 (1975), no. 4,
589–600. MR 0394695
(52 #15494)
 10.
John
Douglas Moore, Conformally flat submanifolds of Euclidean
space, Math. Ann. 225 (1977), no. 1,
89–97. MR
0431046 (55 #4048)
 11.
John
Douglas Moore, Submanifolds of constant positive curvature. I,
Duke Math. J. 44 (1977), no. 2, 449–484. MR 0438256
(55 #11174)
 12.
John
Douglas Moore, Codimension two submanifolds of
positive curvature, Proc. Amer. Math. Soc.
70 (1978), no. 1,
72–74. MR
487560 (80a:53063), http://dx.doi.org/10.1090/S00029939197804875608
 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
(82k:53081)
 14.
John
Douglas Moore, On extendability of isometric immersions of
spheres, Duke Math. J. 85 (1996), no. 3,
685–699. MR 1422362
(97i:53074), http://dx.doi.org/10.1215/S0012709496085269
 15.
M.
S. Narasimhan and S.
Ramanan, Existence of universal connections, Amer. J. Math.
83 (1961), 563–572. MR 0133772
(24 #A3597)
 16.
Barrett
O’Neill, Umbilics of constant curvature immersions, Duke
Math. J. 32 (1965), 149–159. MR 0180951
(31 #5181)
 1.
 I. Amemiya and K. Masuda, On Joristheorem on differentiability of functions, Kodai Math. J. 12 (1989), 9297. MR 90a:26005
 2.
 J. Boman, Differentiability of a function and of its compositions with functions of one variable, Math. Scand. 20 (1967), 249268. MR 38:6009
 3.
 J. Cheeger and J. Simons, Differential characters and geometric invariants, Springer Lecture Notes 1167 (1985), 5080. MR 87g:53059
 4.
 S. S. Chern, A simple intrinsic proof of the GaussBonnet formula for closed Riemannian manifolds, Annals of Math. 45 (1944), 747752. MR 6:106a
 5.
 S. S. Chern and N. Kuiper, Some theorems on the isometric imbedding of compact Riemann manifolds in Euclidean space, Annals of Math. 56 (1952), 422430. MR 14:408e
 6.
 S. S. Chern and J. Simons, Characteristic forms and geometric invariants, Annals of Math. 99 (1974), 4869. MR 50:5811
 7.
 H. Federer, Geometric measure theory, Springer, New York, 1969. MR 41:1976
 8.
 S. Kobayashi and K. Nomizu, Foundations of differential geometry (two volumes), John Wiley and Sons, New York, 1963 and 1969. MR 27:2945; MR 38:6501
 9.
 J. Millson, Examples of nonvanishing ChernSimons invariants, J. Differential Geometry 10 (1975), 589600. MR 52:15494
 10.
 J. D. Moore, Conformally flat submanifolds of Euclidean space, Math. Ann. 225 (1977), 8997. MR 55:4048
 11.
 J. D. Moore, Submanifolds of constant positive curvature I, Duke Math. J. 44 (1977), 449484. MR 55:11174
 12.
 J. D. Moore, Codimension two submanifolds of positive curvature, Proc. Amer. Math. Soc. 70 (1978), 7274. MR 80a:53063
 13.
 J. D. Moore, On conformal immersions of space forms, Springer Lecture Notes 838 (1981), 203210. MR 82k:53081
 14.
 J. D. Moore, On extendability of isometric immersions of spheres, Duke Math. J. 85 (1996), 685699. MR 97i:53074
 15.
 M. S. Narasimhan and S. Ramanan, Existence of universal connections, Amer. J. Math. 83 (1961), 563672. MR 24:A3597
 16.
 B. O'Neill, Umbilics of constant curvature immersions, Duke Math. J. 32 (1965), 8997. MR 31:5181
Similar Articles
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:
http://dx.doi.org/10.1090/S000299470203043X
PII:
S 00029947(02)03043X
Received by editor(s):
March 28, 2001
Published electronically:
May 7, 2002
Article copyright:
© Copyright 2002
American Mathematical Society
