In the present paper we show a model for geodesic flows on the unit tangent bundles of complete Riemannian manifolds. By treating it as in the study of manifolds without conjugate points we have two theorems of the same type as E. Hopf and L. Green proved. One is for spaces of constant curvature instead of flat manifolds. The other is for differentiable flows without conjugate points, and in particular, gradient flows. In addition, we give the formula of the same type as R. Ossermann and P. Sarnak did. As its application, we get the simpler proof of the extension due to W. Ballmann and W. P. Wojtkowski.

We find infinitely many rational solutions of the Ernst equation in general relativity. These are constructed by solving a formal version of Hilbert’s homogeneous problem and are expressed in terms of determinants of Toeplitz type matrices.

We study Gauss maps for submanifolds of riemannian symmetric spaces and show that they have the same properties as the Gauss maps for submanifolds of euclidean spaces.

The gradient flow in a least squares problem on a Lie group takes a Lax form [8]. We associate the Lax equation with homogeneous spaces and symmetric spaces of compact simple Lie groups. The critical points of the Lax equation lie in the Cartan subalgebras of the simple Lie algebras. A reduction from homogeneous spaces to symmetric spaces is described by a ‘coalescence’ of roots. For the complex Grassmann manifold, it is shown that an initial value problem of the Lax equation can be uniquely solved. Some applications to a least squares fitting problem and a linear programming problem are discussed.

The diameter of the set of poles on Riemannian manifolds of nonnegative sectional curvature is estimated by a constant defined by Maeda. We study the constant for elliptic paraboloids and show that our estimate is sharp.