Geodesic
On the connection generated by the problem of minimizing a multiple integral
Sbornik. Mathematics, Tome 188 (1997) no. 1, pp. 61-74 
M. I. Zelikin. On the connection generated by the problem of minimizing a multiple integral. Sbornik. Mathematics, Tome 188 (1997) no. 1, pp. 61-74. http://geodesic.mathdoc.fr/item/SM_1997_188_1_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the Dirichlet integral, a functional which depends quadratically on the sections of a vector bundle $\pi \colon \xi \to \mathfrak N$ over a smooth manifold $\mathfrak N$. We obtain and investigate a quadratic system of partial differential equations, called the Riccati equation by analogy with the one-dimensional case. We show that the solutions of this system define a connection $\nabla$ on the bundle $\xi$. A field of extremals for the Dirichlet functional exists if and only if there is a solution of the Riccati equation that defines a flat connection. The existence of a globally defined solution of the Riccati equation satisfying certain additional conditions guarantees that the Dirichlet functional is positive-definite.

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