On the connection generated by the~problem of minimizing a~multiple integral

M. I. Zelikin

Sbornik. Mathematics, Tome 188 (1997) no. 1, pp. 61-74

Résumé

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.