A note on Morse's index theorem for Perelman's $\mathcal{L}$-length

Details A note on Morse's index theorem for Perelman's $\mathcal{L}$-length A note on Morse's index theorem for Perelman's $\mathcal{L}$-length

2006-02-06

arxiv.org/abs/math/0602090v1

Mathematics

Differential Geometry

4 pages

Scientific paper

This is essentially a note on Section 7 of Perelman's first paper on Ricci flow. We list some basic properties of the index form for Perelman's $ \mathcal{L} $-length, which are analogous to the ones in Riemannian case (with fixed metric), and observe that Morse's index theorem for Perelman's $\mathcal{L}$-length holds. As a corollary we get the finiteness of the number of the $\mathcal{L}$-conjugate points along a finite $\mathcal{L}$-geodesic.

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