L-Geodesic Equation: Full Derivation

The L-Length Functional

Perelman's reduced geometry begins with a length functional adapted to the backward Ricci flow. Let $(M, g(\tau))$ be a solution of the backward Ricci flow $\partial_\tau g = 2\,\mathrm{Ric}$, where $\tau = T - t$ is the backward time parameter. For a path $\gamma: [0, \bar{\tau}] \to M$, define the L-length:

$$\boxed{\mathcal{L}(\gamma) = \int_0^{\bar{\tau}} \sqrt{\tau}\,\bigl(R(\gamma(\tau), \tau) + |\dot{\gamma}(\tau)|^2_{g(\tau)}\bigr)\, d\tau.}$$

The integrand has two terms: the scalar curvature $R$ evaluated along the path, and the kinetic energy $|\dot{\gamma}|^2$ measured in the evolving metric $g(\tau)$. The weight $\sqrt{\tau}$ is chosen so that the resulting Euler-Lagrange equation has remarkable cancellation properties.

Note that unlike a standard Riemannian energy functional, $\mathcal{L}$ depends on the full spacetime geometry through both $R$ and the evolving norm $|\cdot|_{g(\tau)}$.

First Variation with Evolving Metric

Let $\gamma_s$ be a one-parameter family of paths with $\gamma_0 = \gamma$and variation field $Y = \partial_s \gamma_s \big|_{s=0}$. We compute

$$\frac{d}{ds}\bigg|_{s=0} \mathcal{L}(\gamma_s) = \int_0^{\bar{\tau}} \sqrt{\tau}\,\Bigl(\langle \nabla R, Y \rangle + 2\langle \nabla_Y \dot{\gamma}, \dot{\gamma} \rangle + 2\,R_{ij}\,\dot{\gamma}^i Y^j\Bigr)\, d\tau.$$

The third term $2R_{ij}\dot{\gamma}^i Y^j$ arises because the metric is evolving: the variation of $|\dot{\gamma}|^2_{g(\tau)} = g_{ij}(\tau)\dot{\gamma}^i\dot{\gamma}^j$ picks up a contribution from $\partial_\tau g_{ij} = 2R_{ij}$ when the path is varied.

Integration by Parts

In the second term, we swap $\nabla_Y \dot{\gamma} = \nabla_{\dot{\gamma}} Y$ (since$[Y, \dot{\gamma}] = 0$ for a variation) and integrate by parts:

$$\int_0^{\bar{\tau}} 2\sqrt{\tau}\,\langle \nabla_{\dot{\gamma}} Y, \dot{\gamma} \rangle\, d\tau = 2\sqrt{\bar{\tau}}\,\langle Y, \dot{\gamma} \rangle\bigg|_{\bar{\tau}} - \int_0^{\bar{\tau}} \left(\frac{1}{\sqrt{\tau}}\langle Y, \dot{\gamma}\rangle + 2\sqrt{\tau}\,\langle Y, \nabla_{\dot{\gamma}}\dot{\gamma}\rangle\right) d\tau$$

where the boundary term at $\tau = 0$ vanishes because of the $\sqrt{\tau}$ weight. The $1/\sqrt{\tau}$ factor in the second integrand comes from differentiating $\sqrt{\tau}$.

The L-Geodesic Equation

Collecting terms and requiring that the first variation vanishes for all compactly supported $Y$gives the Euler-Lagrange equation:

$$\boxed{\nabla_{\dot{\gamma}}\dot{\gamma} - \frac{1}{2}\nabla R + \frac{1}{2\tau}\dot{\gamma} + 2\,\mathrm{Rc}(\dot{\gamma}, \cdot)^{\sharp} = 0.}$$

This is the L-geodesic equation. Paths satisfying this equation are critical points of $\mathcal{L}$ and are called L-geodesics.

Compared to the standard geodesic equation $\nabla_{\dot{\gamma}}\dot{\gamma} = 0$, there are three correction terms, each with a clear geometric origin.

The Three Correction Terms

1. Curvature Gradient: $-\frac{1}{2}\nabla R$

This term couples the L-geodesic to the scalar curvature of the manifold. It acts as a force that pushes the geodesic toward regions of higher scalar curvature, reflecting the presence of $R$ in the L-length integrand. In flat space ($R = 0$), this term vanishes.

2. Damping Term: $\frac{1}{2\tau}\dot{\gamma}$

This term arises from the $\sqrt{\tau}$ weight in the L-length functional. It acts as a friction or damping force proportional to the velocity, with strength that diverges as $\tau \to 0$. The effect is to slow down L-geodesics near the basepoint $\tau = 0$, analogous to how geodesics in a cone are forced to pass through the tip.

3. Metric Evolution: $2\,\mathrm{Rc}(\dot{\gamma}, \cdot)^{\sharp}$

This term arises from the fact that the metric $g(\tau)$ itself is evolving by $\partial_\tau g = 2\,\mathrm{Ric}$. When computing the variation of the kinetic energy $|\dot{\gamma}|^2_{g(\tau)}$, the time derivative of the metric contributes the Ricci curvature term. This has no analogue in static Riemannian geometry and is unique to the Ricci flow setting.

Reparametrisation and Existence

The substitution $s = 2\sqrt{\tau}$ transforms the L-geodesic equation into a form without the singular $1/(2\tau)$ coefficient. In the new parameter, the equation becomes a standard second-order ODE with smooth coefficients, guaranteeing short-time existence and uniqueness of L-geodesics emanating from any basepoint.

This reparametrisation is also key to Perelman's definition of the reduced distance $\ell(\cdot, \tau) = \mathcal{L}(\gamma_{\min})/(2\sqrt{\tau})$ and the reduced volume $\widetilde{V}(\tau) = \int_M (4\pi\tau)^{-n/2} e^{-\ell}\, d\mu_{g(\tau)}$, whose monotonicity is the central result of Perelman's first paper.

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