Author’s abstract: “We consider the Dirichlet integral, a functional which depends quadratically on the sections of a vector bundle \(\pi :\xi\rightarrow \mathfrak N\) 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”.