The Morse index theorem where the ends are submanifolds
HTML articles powered by AMS MathViewer
- by Diane Kalish PDF
- Trans. Amer. Math. Soc. 308 (1988), 341-348 Request permission
Abstract:
In this paper the Morse Index Theorem is proven in the case where submanifolds $P$ and $Q$ are at the endpoints of a geodesic, $\gamma$. At $\gamma$, the index of the Hessian of the energy function defined on paths joining $P$ and $Q$ is computed using $P$-focal points, and a calculation at the endpoint of $\gamma$, involving the second fundamental form of $Q$.References
- W. Ambrose, The index theorem in Riemannian geometry, Ann. of Math. (2) 73 (1961), 49–86. MR 133783, DOI 10.2307/1970282
- John Bolton, The Morse index theorem in the case of two variable end-points, J. Differential Geometry 12 (1977), no. 4, 567–581 (1978). MR 512926
- Richard L. Bishop and Richard J. Crittenden, Geometry of manifolds, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. MR 0169148
- J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 308 (1988), 341-348
- MSC: Primary 58E10
- DOI: https://doi.org/10.1090/S0002-9947-1988-0946447-5
- MathSciNet review: 946447