  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.
