Zermelo deformation of Finsler metrics by Killing vector fields

Let $\gamma(t)$ be an arc-length-parameterized $F$-geodesic, we need to prove that the curve $t\mapsto\Psi_{t}(\gamma(t))$ is an arc-length-parameterized $\tilde{F}$-geodesic. In order to do it, observe that for any $F$-arc-length-parameterized curve $x(t)$ the $t$-derivative of $\Psi_{t}(x(t))$ is given by $\Psi_{t*}(\dot{x}(t))+v(\Psi_{t}(x(t)))$. Since the flow $\Psi_{t}$ preserves $F$ and $v$, it preserves $\tilde{F}$ and therefore

The last equality in the formula above is true because, for any $\xi$ such that $F(x,\xi)=1$, we have $\tilde{F}\left(x,\xi+v(x)\right)=F(x,\xi)$ by the definition of the Zermelo deformation. Thus, if the curve $x(t)$ is $F$-arc-length-parameterized, then the curve $\Psi_{t}(x(t))$ is $\tilde{F}$-arc-length-parameterized.

This also implies that the integrals $\int F(x(t),\dot{x}(t))dt$ and $\int\tilde{F}\left(\Psi_{t}(x(t)),\dot{\left(\Psi_{t}(x(t))\right)}\right)dt$ coincide for all $F$-arc-length-parameterized curves $x(t)$. Since geodesics are locally the shortest arc-length parameterized curves connecting two points, for each arc-length parameterized $F$-geodesic $\gamma$ the curve $t\mapsto\Psi_{t}(\gamma(t))$ is a $\tilde{F}$-arc-length-parameterized geodesic as we claimed.

Let us now prove the second statement of Theorem 1. Consider a Jacobi vector field $J(t)$ which are orthogonal to $\gamma$. We need to show that the pushforward $\tilde{J}(t)=\Psi_{t*}(J(t))$ is a Jacobi vector field for the $\tilde{F}$-geodesic $\tilde{\gamma}(t):=\Psi_{t}(\gamma(t))$. By the definition of Jacobi vector field there exists a family $\gamma_{s}(t)$ of geodesics with $\gamma_{0}=\gamma$ such that $J(t)=\frac{d}{ds}_{|s=0}\gamma_{s}(t)$, since $J(t)$ is orthogonal to $\gamma$ we may assume that all geodesics $\gamma_{s}(t)$ are arc-length parameterized. As we explained above, $\Psi_{t}(\gamma_{s}(t))$ is a family of $\tilde{F}$-geodesics; taking the derivative by $s$ at $s=0$ proves what we want.

Let us now show that $\tilde{J}$ is orthogonal to $\dot{\tilde{\gamma}}$. First observe that the condition that $J(t)$ is orthogonal to $\dot{\gamma}(t)$ is equivalent to the condition that $J(F):=\sum_{r}J^{r}\frac{\partial F}{\partial\xi_{r}}$ vanishes at $\dot{\gamma}(t)$ for each $t$. Indeed, consider the one-form $U\in T_{\gamma(t)}M^{n}\mapsto g_{\gamma(t),\dot{\gamma}(t)}(\dot{\gamma}(t)_{,}U)$. Because of the (positive) homogeneity of the function $F$ we have that at a point $\dot{\gamma}(t)\in T_{\gamma(t)}M^{n}$

Next, take Equation (1) and calculate the differential of the restriction of $\tilde{F}$ to the tangent space: its components are given by

In this formula, the derivatives of the function $F$ are taken at $\xi\in T_{x}S^{2}$, and the derivatives of the function $\tilde{F}$ are taken at $\xi-\tilde{F}(x,\xi)v$. By $v(F)$ we denoted the function $\sum_{r}\frac{\partial F}{\partial\xi_{r}}v^{r}$.

In view of (5), $J(\tilde{F}):=\sum_{r}J^{r}\frac{\partial\tilde{F}}{\partial\xi_{r}}$ vanishes at $\dot{\gamma}(t)+v(\gamma(t))$, so $J(t)$ is orthogonal to $\dot{\gamma}(t)+v(\gamma(t))$ (the orthogonality is understood in the sense of $\tilde{g}_{(\gamma(t),\dot{\gamma}(t)+v(\gamma(t)))}$). Then, $\tilde{J}(t)=\Psi_{t*}(J(t))$ is $\tilde{g}_{(\tilde{\gamma}(t),\dot{\tilde{\gamma}}(t))}$ orthogonal to $\dot{\tilde{\gamma}}(t).$

Let us now prove the third statement of Theorem 1, we need to show that $K(x,\xi,\eta)=\tilde{K}(x,\xi+v,\eta)$. We consider a $F$-geodesic $\gamma(t)$ with $\gamma(0)=0$ and $\dot{\gamma}(0)=\xi$ and the corresponding $\tilde{F}$-geodesic $\tilde{\gamma}(t)=\Psi_{t}(\gamma(t));$ since $\Psi_{0}=\textrm{Id}$, we have $\dot{\tilde{\gamma}}(0)=\dot{\gamma}(0)+v=\xi+v$.

Observe that by combining [2, Eqn. 6.16 on page 117] and [2, Eqn. 6.3 on page 108] we obtain

We will assume that $J(0):=\eta$ is $g$-orthogonal to $\dot{\gamma}(0)$. As explained above this implies that $\dot{\tilde{\gamma}}(0)$ is $\tilde{g}$-orthogonal to $\tilde{J}(0)$. Then, by (6), the minimum of $\frac{d^{2}}{dt^{2}}g(J(t),J(t))_{|t=0}$ taken over all Jacobi vector fields $J$ along $\gamma$ which are equal to $\eta$ at $t=0$, is equal to $-K(x,\xi,\eta)g(\xi,\xi)g(\eta,\eta)$. An analogous statement is clearly true also for $\tilde{\gamma}$, $\tilde{g}$ and $\tilde{J}$: namely, the minimum of $\frac{d^{2}}{dt^{2}}\tilde{g}(\tilde{J}(t),\tilde{J}(t))_{|t=0}$ taken over all Jacobi vector fields $\tilde{J}$ along $\tilde{\gamma}$ which are equal to $\eta$ at $t=0$, is equal to $-\tilde{K}(x,\xi+v,\eta)\tilde{g}(\xi+v,\xi+v)\tilde{g}(\eta,\eta)$. Here we used the relation $\tilde{J}(0)=\Psi_{t*}(J(t))_{|t=0}=J(0)$. Finally, in order to show that $K(x,\xi,\eta)=\tilde{K}(x,\xi+v,\eta),$ it is sufficient to show that the function $t\mapsto g(J(t),J(t))$ is proportional, with a constant coefficient, to the function $t\mapsto\tilde{g}(\tilde{J}(t),\tilde{J}(t)).$

In order to prove this, let us first compare $g_{(\gamma(t),\dot{\gamma}(t))}=\frac{1}{2}d_{\xi}^{2}F^{2}_{(\gamma(t),\dot{\gamma}(t))}$ and

It is convenient to work in coordinates $(x_{1},...,x_{n})$ such that the entries of $v$ are constants, in these coordinates for each $t$ the differential of the diffeomorphism $\Psi_{t}$ is given by the identity matrix, so in these coordinates $J(t)=\tilde{J}(t)$ and $\dot{\tilde{\gamma}}(t)=\dot{\gamma}(t)+v(\gamma(t)).$

Differentiating (5), we get the second derivatives of $\tilde{F}$. They are given by

Again, all derivatives of the function $F$ are taken at $\xi$, and of the function $\tilde{F}$ are taken at $\xi-\tilde{F}(x,\xi)v(x)$. Note that one term in the brackets in (7) appears because we differentiate $\frac{1}{1+v(F)}$, and the other appears because the derivatives of $\frac{\partial F}{\partial\xi_{j}}$ are taken at $\xi-\tilde{F}(x,\xi)v(x)$. When we differentiate it, we also need to take into account the additional term $-\tilde{F}(x,\xi)v(x)$.

Now, in view of the formula $\frac{1}{2}d^{2}(\tilde{F}^{2})=\tilde{F}d^{2}\tilde{F}+d\tilde{F}\otimes d\tilde{F}$ we obtain from (7) the formula for $\tilde{g}_{ij}$:

Let us now compare the length of $J$ in $g(x,\xi)$ with that of in $\tilde{g}(x,\xi+v)$. We multiply (8) by $J^{i}J^{j}$ and sum with respect to $i$ and $j$. Since by assumptions $J(F)=\sum_{r}J^{r}\frac{\partial F}{\partial\xi_{r}}$ vanishes at $\xi$, all terms in the sum but the first vanish. We thus obtain that the length of $J$ in $\tilde{g}$ is proportional to that of in $g$ with the coefficient which is the square root of $\frac{\tilde{F}}{1+v(F)}$.

But along the geodesic both $\tilde{F}$ and $v(F)$ are constant. Indeed, $v(F)$ is the “Noether” integral corresponding to the Killing vector field. Theorem 1 is proved.

First, observe that a Finsler metric is locally symmetric if and only if for any geodesic $\gamma$ and any Jacobi vector field $J$ along $\gamma$ the vector field $D_{\dot{\gamma}}J$ is also a Jacobi vector field. Indeed, the equation for Jacobi vector fields is

$D_{\dot{\gamma}}$-differentiating this equation, we obtain

If $D_{\dot{\gamma}}J$ is a Jacobi vector field, $D_{\dot{\gamma}}D_{\dot{\gamma}}(D_{\dot{\gamma}}J)+R_{\dot{\gamma}}(D_{\dot{\gamma}}J)$ vanishes so the equation above implies $\left(D_{\dot{\gamma}}R_{\dot{\gamma}}\right)(J)=0$, and since it is fulfilled for all Jacobi vector fields we have $D_{\dot{\gamma}}R_{\dot{\gamma}}=0$ as we claimed.

Thus, we assume that for any geodesic and for any Jacobi vector field for $F$ its $D_{\dot{\gamma}}$ derivative is also a Jacobi vector field, and our goal is to show the same for $\tilde{F}$. Clearly, it is sufficient to show this only for Jacobi vector fields which are $g$-orthogonal to $\dot{\gamma}$. Note that for such Jacobi vector fields $D_{\dot{\gamma}}J$ is also orthogonal to $\dot{\gamma}$, since both $g_{(\gamma,\dot{\gamma})}$ and $\dot{\gamma}$ are $D_{\dot{\gamma}}$-parallel.

Take a (arc-length-parameterized) $F$-geodesic $\gamma$ and a point $P=\gamma(0)$ on it. Consider the geodesic polar coordinated around this point, let us recall what they are and their properties which we use in the proof.

Consider the (local) diffeomorphism of $T_{P}M^{n}\setminus\{0\}$ to $M^{n}$ which sends $\xi\in T_{P}M^{n}\setminus\{0\}$ to $\exp(\xi):=\gamma_{\xi}(1)$, where $\gamma_{\xi}$ is the geodesic starting from $P$ with the velocity vector $\xi$. As the local coordinate systems on $T_{P}M^{n}\setminus\{0\}$ we take the following one: we choose a local coordinate system $x_{1},...,x_{n-1}$ on the unit $F$-sphere $\{\xi\in T_{P}M^{n}\mid F(P,\xi)=1\}$ and set the tuple $\left(F(P,\xi),x_{1}\left(\frac{1}{F}\xi\right),...,x_{n-1}\left(\frac{1}{F}\xi\right)\right)$ to be the coordinates of $\xi$. Combining it with the diffeomorphism $\exp$, we obtain a local coordinate system on $M^{n}$. By construction, in this coordinate system each arc-length parameterized geodesics starting at $P$, in particular the geodesic $\gamma$, is a curve of the form $(t,\textrm{const}_{1},...,\textrm{const}_{n-1})$.

Next, consider the following local Riemannian metric $\hat{g}$ in a punctured neighborhood of $P$: for a point $\sigma(t)$ of this neighborhood such that $\sigma$ is a geodesic passing through $P$ we set $\hat{g}:=g_{(\sigma(t),\dot{\sigma}(t))}$.

It is known that in the polar coordinates the metric $\hat{g}$ is block-diagonal with one $1\times 1$ block which is simply the identity and one $(n-1)\times(n-1)$-block which we denote by $G$:

It is known, see e.g. [9, Lemma 7.1.4], that the geodesics passing through $P$ are also geodesics of $\hat{g}$, and that for each such geodesic the operator $\hat{\nabla}_{\dot{\gamma}}$, where $\hat{\nabla}$ is the Levi-Civita connection of $\hat{g}$, coincides with $D_{\dot{\gamma}}$.

Next, consider analogous objects for the metric $\tilde{F}$. As the local coordinate system on the unit $\tilde{F}$-sphere we take the following: as the coordinate tuple of $\tilde{\xi}$ with $\tilde{F}(P,\tilde{\xi})=1$ we take the coordinate tuple $(x_{1}(\xi),...,x_{n-1}(\xi))$, where $\xi:=\tilde{\xi}-v(P)$. (Recall that the $v$-parallel-transport sends the unit $F$-sphere to the unit $\tilde{F}$-sphere.)

By Theorem 1, in these coordinate systems each Jacobi vector field $J=(J_{0}(t),...,J_{n-1}(t))$ along $\gamma$ which is orthogonal to $\gamma$ is also a Jacobi vector field along $\tilde{\gamma}$, which is the $\tilde{F}$-geodesic such that $\tilde{\gamma}(0)=P$ and $\dot{\tilde{\gamma}}(0)=\dot{\gamma}(0)+v(P)$, and is orthogonal to $\tilde{\gamma}$. By (8), the corresponding block $\tilde{G}$ is given by $\tilde{G}=\frac{1}{1+v(F)}G$. Since the function $v(F)$ is constant along geodesics, the coefficients $\Gamma_{ij}^{k}$ of the Levi-Civita connection $\hat{\nabla}$ of $\hat{g}$ such that $i=0$ or $j=0$ coincide with that of for the analog for $\tilde{F}$. A direct way to see the last claim is to use the formula $\Gamma_{ij}^{k}=\frac{1}{2}g^{ks}\left(\frac{\partial g_{is}}{\partial x_{j}}+\frac{\partial g_{js}}{\partial x_{i}}-\frac{\partial g_{ij}}{\partial x_{s}}\right)$, where of course all indices run from $0$ to $n-1$ and the summation convention is assumed.

Then, in our chosen coordinate system, the formula for the covariant derivative in $\hat{\nabla}$ along $\gamma$ for vector fields which are orthogonal to $\gamma$ simply coincides with that of the formula for the corresponding objects for $\tilde{F}$. Then, for any $\tilde{F}$-Jacobi vector field $\tilde{J}$ orthogonal to $\dot{\tilde{\gamma}}$ we have that $\tilde{D}_{\dot{\tilde{\gamma}}}\tilde{J}$- is again a Jacobi vector field. Theorem 2 is proved.

Acknowledgments. The authors thank Sergei Ivanov for useful comments. V.M. was partially supported by the University of Jena and by the DFG grant MA 2565/4.