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Conjugate points on a limiting extremal.

M Morse1

  • 1Institute for Advanced Study, Princeton, New Jersey 08540.

Proceedings of the National Academy of Sciences of the United States of America
|June 1, 1976
PubMed
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Nondegenerate extremals on Riemannian manifolds, when close to a given extremal, maintain their nondegeneracy and index. This is crucial for understanding Weierstrass Integrals and variational analysis.

Area of Science:

  • Differential Geometry
  • Global Analysis
  • Calculus of Variations

Background:

  • Weierstrass Integrals are fundamental in the calculus of variations, particularly on Riemannian manifolds.
  • Understanding the properties of extremals, such as degeneracy and index, is key to analyzing these integrals.
  • Previous work established foundational theorems on Riemannian manifolds.

Purpose of the Study:

  • To establish Theorem 27.3 from Global Variational Analysis using extensions of Sturmian Comparison Theorems.
  • To analyze the properties of nondegenerate extremals of a Weierstrass Integral on a compact, connected, Riemannian manifold.
  • To demonstrate how nearby extremals inherit properties from a given nondegenerate extremal.

Main Methods:

  • Utilizing Theorem 1 and extensions of the Sturmian Comparison Theorems.

Related Experiment Videos

  • Analyzing a Weierstrass Integral on a compact, connected, Riemannian manifold M(n).
  • Defining and counting conjugate points to determine the index of an extremal.

    • Main Results:

    • Theorem 1 states that nondegenerate extremals near a given extremal gamma, with the same length and in the Fréchet sense, remain nondegenerate.
    • These nearby extremals also inherit the same index as the original extremal gamma.
    • This result supports the establishment of Theorem 27.3 in Global Variational Analysis.

    Conclusions:

    • The nondegeneracy and index of extremals are stable properties under small perturbations on Riemannian manifolds.
    • This stability is essential for the rigorous analysis of Weierstrass Integrals.
    • The findings contribute to a deeper understanding of global variational analysis on manifolds.