Let U⊂Rn be an open set, and let L:U×Rn→R be a continuous, in its second variable first order positive homogeneous, positive over U×(Rn∖{0}) and of C2-class Langrange function, such that for all p∈U the Gauss-curvature of the hyper surface
\{ v\in\mathbb R^n \mid L(p,v) \equal{} 1 \}
is nowhere zero. Determine the extremals of L if it satisfies the following system
\sum_{k \equal{} 1}^n y^k\partial_k\partial_{n \plus{} i}L \equal{} \sum_{k \equal{} 1}^n y^k\partial_i\partial_{n \plus{} k} L \qquad (i\in\{1,\dots,n\})
of partial differetial equations, where y^k(u,v) : \equal{} v^k for (u,v)∈U×Rk, v \equal{} (v^1,\dots,v^k). functionGausstopologyadvanced fieldsadvanced fields unsolveddifferential geometry