Homotopy type of path spaces
Author:
Diane Kalish
Journal:
Proc. Amer. Math. Soc. 116 (1992), 259271
MSC:
Primary 58B05; Secondary 55P99, 58E05, 58E10
Erratum:
Proc. Amer. Math. Soc. 118 (1993), null.
MathSciNet review:
1097347
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: This paper extends the Fundamental Theorem of Morse Theory to the two endmanifold case. The theorem relates the homotopy type of the space of paths connecting two submanifolds of a Riemannian manifold to the critical points of the energy function defined on this path space. Use of the author's formulation of the Morse index Theorem in this setting allows for a simple computation of the homotopy type, and several specific examples are worked out.
 [1]
W.
Ambrose, The index theorem in Riemannian geometry, Ann. of
Math. (2) 73 (1961), 49–86. MR 0133783
(24 #A3608)
 [2]
Richard
L. Bishop and Richard
J. Crittenden, Geometry of manifolds, Pure and Applied
Mathematics, Vol. XV, Academic Press, New YorkLondon, 1964. MR 0169148
(29 #6401)
 [3]
John
Bolton, The Morse index theorem in the case of two variable
endpoints, J. Differential Geom. 12 (1977),
no. 4, 567–581 (1978). MR 512926
(80b:58025)
 [4]
D. Kalish, Aspects of Morse theory, doctoral dissertation, CUNY, 1984.
 [5]
Diane
Kalish, The Morse index theorem where the ends
are submanifolds, Trans. Amer. Math. Soc.
308 (1988), no. 1,
341–348. MR
946447 (89i:58024), http://dx.doi.org/10.1090/S00029947198809464475
 [6]
J.
Milnor, Morse theory, Based on lecture notes by M. Spivak and
R. Wells. Annals of Mathematics Studies, No. 51, Princeton University
Press, Princeton, N.J., 1963. MR 0163331
(29 #634)
 [7]
Marston
Morse, The calculus of variations in the large, American
Mathematical Society Colloquium Publications, vol. 18, American
Mathematical Society, Providence, RI, 1996. Reprint of the 1932 original.
MR
1451874 (98f:58070)
 [1]
 W. Ambrose, The index theorem in Riemannian geometry, Ann. of Math. (2) 73 (1961), 4986. MR 0133783 (24:A3608)
 [2]
 R. Bishop and R. Crittenden, Geometry of manifolds, Pure Appl. Math., vol. 15, Academic Press, New York, 1964. MR 0169148 (29:6401)
 [3]
 J. Bolton, The Morse index theorem in the case of two variables endpoints, J. Differential Geom. 12 (1977), 567581. MR 512926 (80b:58025)
 [4]
 D. Kalish, Aspects of Morse theory, doctoral dissertation, CUNY, 1984.
 [5]
 , The Morse index theorem where the ends are submanifolds, Trans. Amer. Math. Soc.. 308 (1988), 341348. MR 946447 (89i:58024)
 [6]
 J. Milnor, Morse theory, Ann. of Math. Stud., no. 51, Princeton Univ. Press, Princeton, NJ, 1973. MR 0163331 (29:634)
 [7]
 M. Morse, The calculus of variations in the large, Amer. Math. Soc. Colloq. Publ., vol. 18, Amer. Math. Soc., Providence, RI, 1934. MR 1451874 (98f:58070)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
58B05,
55P99,
58E05,
58E10
Retrieve articles in all journals
with MSC:
58B05,
55P99,
58E05,
58E10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199210973470
PII:
S 00029939(1992)10973470
Article copyright:
© Copyright 1992
American Mathematical Society
