The aim of this paper is to give an algebraic independence result for the two infinite products involving the Lucas sequences of the first and second kind. As a consequence, we derive that the two infinite products ∏_k=1^∞(1+1/F_2^k) and ∏_k=1^∞(1+1/L_2^k) are algebraically independent over ℚ, where {F_n}_n≥0 and {L_n}_n≥0 are the Fibonacci sequence and its Lucas companion, respectively.

We give examples of certain kind of minimal orbits of Hermann actions and discuss whether each of the examples is austere.

A univalence condition for certain class of analytic functions was discussed by D. Yang and S. Owa (Hokkaido Math. J. 32 (2003), 127-136). In the present paper, by discussing some subordination relation, a new univalence condition is deduced.

Asmar, Montgomery-Smith and Saeki gave a generalization of a theorem of Bochner for a locally compact abelian group with certain direction. We show that a strong version of their result holds for a σ-compact, connected locally compact abelian group with certain direction. We also give several conditions for quasi-invariance of analytic measures and another proof of a theorem of deLeeuw and Glicksberg.

We shall show that there are infinite pairs of non-direct product 2-groups with the same character. They are not pairs of the generalized quaternion group and dihedral group.

In this paper, the formulation of the biharmonic map equation in terms of the Maurer-Cartan form for all smooth maps of a compact Riemannian manifold into a compact Lie group (G,h) with the bi-invariant Riemannian metric h is obtained. Using this, all biharmonic curves into compact Lie groups are determined exactly, and all the biharmonic maps of an open domain of ℝ² equipped with a Riemannian metric conformal to the standard Euclidean metric into (G,h) are determined.

In this paper, the formulation of the biharmonic map equation in terms of the Maurer-Cartan form for all smooth maps of a compact Riemannian manifold into a Riemannian symmetric space (G/K,h) induced from the bi-invariant Riemannian metric h on G is obtained. Using this, all the biharmonic curves into symmetric spaces are determined, and all the biharmonic maps of an open domain of ℝ² with the standard Riemannian metric into (G/K,h) are characterized exactly.

In this paper we study congruency of minimal ruled real hypersurfaces in a nonflat complex space form with respect to the action of its isometry group. We show that those in a complex hyperbolic space are classified into 3 classes and show that those in a complex projective space are congruent to each other hence form just one class.