One of the basic problems in the theory of dynamical systems is the characterization of stable systems.

Let M be a closed (i.e. compact without boundary) connected smooth manifold with a smooth Riemannian metric and Diffr (M) (r ≥ 1) denote the space of Cr diffeomorphisms on M with the uniform Cr topology.

We denote by Pn(q) the projective space of dimension n over a finite field GF(q) with q elements, and we mean by an i-flat a linear subspace of dimension i in Pn(q).

We consider a compact smooth manifold M. Diff1 (M) denotes the space of C1-diffeomorphisms of M onto itself with the usual C1-topology. In the research of the qualitative theory of dynamical systems there is a desire to find a concept of stability of geometric global structure of orbits such that this stable systems are dense in the space of dynamical systems on M. Structural stability does not satisfy the density condition in Diff1 (M). Tolerance stability (see Section 2 for definition) is a candidate for the density property [7, p. 294]. Concerning tolerance stability there are researches as [6], [7], [8], and [2].

One finds the following example in [3, (34, B)]:

Let k be a field of characteristic p and be n-variables over k. Then if p > 0 and [k : kp] = ∞, is an n-dimensional regular local ring but not a Nagata ring. In particular it is not an excellent ring.

Complete parallel submanifolds of a real space form of constant sectional curvature k have been completely classified by Ferus [3] when k ≧ 0, and by Takeuchi [19] when k < 0. A complex space form is by definition a 2n-dimensional simply connected Hermitian symmetric space of constant holomorphic sectional curvature c and will be denoted by (c).

In [10] A. Morimoto proved that every topologically stable homeomorphism of a compact manifold M has the pseudo-orbit tracing property in the case dim (M) ≥ 2. Further, in studying relation between the topological stability and other stability of diffeomorphisms, he showed the following

Theorem A. Let Rr be the r-dimensional vector group and φ be a group automorphism of Rr.

When he developed the theory of central extensions of absolute abelian fields in [1], Fröhlich clearly pointed out a role of Schur’s multiplicators of the Galois groups in algebraic number theory. Another role of them was to be well known when the gaps between the everywhere local norms and the global norms of finite Galois extensions were cohomologically described by Tate [10]. The relation of two roles was investigated by Furuta [2], Shirai [9], Heider [3] and others.

Let M be a smooth compact manifold and let be a smooth codimension q Riemannian foliation of M. Let T(M) be the tangent bundle of M and let E ⊂ T(M) be the subbundle tangent to . We may regard the normal bundle Q = T(M)/E of as a subbundle of T(M) satisfying T(M) = E ⊕ Q. Let g be a smooth Riemannian metric on Q invariant under the natural parallelism along the leaves of .

The purpose of this paper is to define minimality of surfaces in an infinite dimensional space E by probabilistic methods with the description of the relation between minimal surfaces and harmonic functions on the space E, and to analyze purely analytic properties of a certain class of quadratic forms on the space E.