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- W2024532817 abstract "Let M be a complete Riemann manifold with dimension m and metric g. For p, q ∈ M and ℓ > 0, let the index I (g, p, q, ℓ) be the number of g-geodesics of length ℓ that join p to q. The following generic bounds for this index are the main results we present here. We denote by ℛ the space of complete Riemann metrics on M.(a) For each p ∈ M, there is a residual 𝒢 (p) ⊂ ℛ such that for all g ∈ 𝒢(p)(b) If M is compact, there is a residual 𝒢 ⊂ ℛ such that for all g ∈ 𝒢These finiteness results are part of our study of the focal decomposition—i.e., the partitionStability of this focal deomposition (as g varies) has a natural meaning, in analogy with structural stability in the theory of dynamical systems, and here we begin an investigation in that direction. Our methods involve the multi-transversality theory of J. Mather and the Bumpy Metric Theorem of R. Abraham, as proved by D. Anosov." @default.
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- W2024532817 date "2006-01-25" @default.
- W2024532817 modified "2023-10-16" @default.
- W2024532817 title "Focal stability of Riemann metrics" @default.
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- W2024532817 doi "https://doi.org/10.1515/crelle.2006.029" @default.
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