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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] with [[tangent bundle|WY6K3P4YCG]] $TM$, and write $0$ for the zero section.\nA *Finsler function* on $M$ is a continuous function $F \\colon TM \\to [0, \\infty)$ such that:\n\n1. $F$ is smooth on $TM \\setminus 0$;\n2. $F$ is positively homogeneous: $F(\\lambda v) = \\lambda F(v)$ for all $v \\in TM$ and all $\\lambda \u003e 0$;\n3. $F$ is strongly convex: for each $p \\in M$ and nonzero $v \\in T_pM$, the [[Hessian|0SCSQWXRA4]] at $v$ of $\\tfrac{1}{2} F^2$ on $T_pM$ is [[positive definite|0QQCG3JWJ4]].\n\nA *Finsler manifold* is a pair $(M, F)$ of a smooth manifold and a Finsler function on it.\n\n## Remarks\n\nExample: if $g$ is a Riemannian metric on $M$, then $F(v) = \\sqrt{g(v, v)}$ is a Finsler function.\nIt is smooth away from $0$, homogeneous of degree $1$, and the Hessian of $\\tfrac{1}{2}F^2 = \\tfrac{1}{2} g(v, v)$ on $T_pM$ is $g_p$ itself, which is positive definite.\n\nNon-example: on $\\R^2$, $F(x, v) = \\abs{v_1} + \\abs{v_2}$ is positively homogeneous, but it is not smooth along the lines $v_1 = 0$ and $v_2 = 0$, so clause 1 fails.\n\nClauses 2 and 3 imply that $F(v) \u003e 0$ for $v \\ne 0$ and that $F$ satisfies the triangle inequality on each tangent space.\nThe function $F$ need not be symmetric: $F(-v) = F(v)$ is not required, and Randers metrics are standard examples with $F(-v) \\ne F(v)$.\nThe length of a piecewise smooth curve $\\gamma$ is $\\int F(\\gamma'(t)) \\, dt$.\n",
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