\u6211\u4eec\u77e5\u9053, \u5728\u9ece\u66fc\u51e0\u4f55\u4e2d, \u5173\u4e8e\u5e38\u622a\u9762\u66f2\u7387\u7684\u5b8c\u5907\u9ece\u66fc\u6d41\u5f62(\u5373\u7a7a\u95f4\u5f62\u5f0f)\u7684\u5206\u7c7b\u5df2\u7ecf\u5b8c\u5168\u89e3\u51b3:<\/p>\n
\\begin{thm} 2004\u5e74, D, Bao, C, Robles & Z, Shen \u8bc1\u660e\u4e86\u5173\u4e8e\u5e38\u65d7\u66f2\u7387\u7684Randers\u5ea6\u91cf\u7684\u5206\u7c7b\u5b9a\u7406, \u8bc1\u660e\u4e3b\u8981\u4f9d\u8d56\u4e8e\u5982\u4e0b\u7ed3\u679c: note that the first equatiuon of \\eqref{eq:1} implies that $v$ is homothetic with respect to $h$, i.e., $L_vg=-2c g$, thus, we can solve it for any given $c$ (in the case that $h$ is of constant sectional curvature $\\mu$).<\/p>\n Given a Finsler metric $F$ and a vector field $v$ with $F(x,v_x)<1$, we can define a new Finsler metric $\\tilde F$ by \u540e\u6765, Robles \u57282007\u5e74\u5f97\u5230\u4e86\u5e38\u65d7\u66f2\u7387\u7684Randers\u5ea6\u91cf\u7684\u6240\u6709\u6d4b\u5730\u7ebf. \\begin{thm} \u6211\u4eec\u77e5\u9053, \u5728\u9ece\u66fc\u51e0\u4f55\u4e2d, \u5173\u4e8e\u5e38\u622a\u9762\u66f2\u7387\u7684\u5b8c\u5907\u9ece\u66fc\u6d41\u5f62(\u5373\u7a7a\u95f4\u5f62\u5f0f)\u7684\u5206\u7c7b\u5df2\u7ecf …<\/p>\n
\n\u5bf9\u6bcf\u4e2a$c\\in\\R$\u4ee5\u53ca\u6240\u6709\u7684$n\\in\\Z^+$, \u90fd\u5b58\u5728\u552f\u4e00\u7684(\u53ea\u76f8\u5dee\u4e00\u4e2a\u7b49\u8ddd)\u7684\u5355\u8fde\u901a\u7684$n$\u4e3a\u7a7a\u95f4\u5f62\u5f0f, \u4f7f\u5f97\u5176\u5e38\u622a\u9762\u66f2\u7387\u4e3a$c$.
\n\\end{thm}
\n\\begin{proof}
\nc.f. [\u4f0d\u9e3f\u71991989<\/a>] P70 Thm1 \u4e0e P97 Thm10.
\n\\end{proof}<\/p>\n
\n
\n\\begin{thm}
\nLet $F$ be a Randers metric and $(h,v)$ be its navigation representation. Then $F$ has constant flag curvature iff $(h,v)$ satisfies:
\n$$
\n\\begin{cases}
\nv_{i|j}+v_{j|i}=-4c h_{ij}\\tag{1}\\\\
\nK=\\mu-c^2
\n\\end{cases}
\n$$
\nwhere $\\mu$ is the constant sectional curvature of the Riemannian metric $h$, $K$ is the constant flag curvature of $F$, and $c$ is a constant.
\n\\end{thm}<\/p>\nNavigation Problem<\/h4>\n
\n\\begin{equation}
\nF(x,y\/\\tilde F(x,y)+v_x)=1.
\n\\end{equation}
\n\\begin{thm}[Chern-Shen, Lemma 1.4.1] For any Piecewise $C^\\infty$ curve $c$ in $M$, the $\\tilde F$-length of $c$ is equal to the time for which the object travels along $c$.
\n\\end{thm}
\n\u7279\u522b, \u5f53$F$\u662f\u4e00\u4e2a\u9ece\u66fc\u5ea6\u91cf\u65f6, \u53ef\u4ee5\u8bc1\u660e\u5bfc\u822a\u95ee\u9898\u7684\u89e3 $\\tilde F$ \u662f\u4e00\u4e2aRanders\u5ea6\u91cf, \u800c\u4e14\u53cd\u8fc7\u6765\u4e5f\u662f\u5bf9\u7684. \u5373, \u7ed9\u5b9a\u4e00\u4e2aRanders\u5ea6\u91cf, \u5b83\u53ef\u4ee5\u770b\u51fa\u5728\u67d0\u4e2a\u9ece\u66fc\u5ea6\u91cf\u4e0b\u5bfc\u822a\u95ee\u9898\u7684\u89e3. \u5177\u4f53\u7684\u8bc1\u660e\u53ef\u4ee5\u53c2\u8003\u6211\u7684\u7b14\u8bb0:Notes on Riemannian-Finsler Geometry<\/em> (Prop 3.1)<\/strong><\/a>.<\/p>\n
\n\\begin{defn}
\nA smooth curve in a Finsler manifold is called a geodesic if it is locally the shortest path connecting two points on this curve.
\n\\end{defn}<\/p>\n
\n[C. Robles, 2007] Let $F$ be a Randers metric of constant flag curvature and $(h,v)$ be its navigational representation, then the geodesic of $F$ are given by $\\psi_t(\\gamma(a(t)))$, where $\\psi_t$ is the flow of $-v$ and $\\gamma(t)$ is a geodesic of $h$ and $a(t)$ is defined by
\n$$
\na(t)=\\begin{cases}
\n\\frac{e^{2ct}-1}{2c},&c\\neq0\\
\nt,&c=0.
\n\\end{cases}
\n$$
\n\\end{thm}<\/p>\nReferences<\/h4>\n
\n