We study lengths of helices of orders 3 and 4 which are generated by some Killing vector fields on a complex projective plane and on a complex hyperbolic plane. We consider the moduli space of such helices under the congruence relation and give a lamination structure on this space which are closely related with the length spectrum. This shows that the moduli space does not form a canonical building structure with respect to the length spectrum.

Let p be a prime not equal to 2 or 3. We determine the group of all modular units on the modular curve X₁(2p), and its full cuspidal class number. We mention a fact concerning the non-existence of torsion points of order 5 or 7 of elliptic curves over Q of square-free conductor n as an application of a result by Agashe and the cuspidal class number formula for X₀(n). We also state the formula for the order of the subgroup of the Q-rational torsion subgroup of J₁(2p) generated by the Q-rational cuspidal divisors of degree 0.

We correct a theorem on the conductor of elliptic curves over Q given in Introduction of the paper “The cuspidal class number formula for the modular curves X₁(2p)”.

For a finite group G, an L(G)-free gap G-module V is a finite dimensional real G-representation space satisfying the two conditions: (1) V^L = 0 for any normal subgroup L of G with prime power index. (2) dim V^P > 2 dim V^H for any P < H ≤ G such that P is of prime power order. A finite group G not of prime power order is called a gap group if there is an L(G)-free gap G-module. We give a necessary and sufficient condition for that G is a gap group for a finite group G satisfying that G/[G,G] is not a 2-group, where [G,G] is the commutator subgroup of G.

We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces, prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces, and prove the two-dimensional pseudo-Riemannian version of the projective Obata conjecture.

Given a strictly pseudoconvex hypersurface M ⊂ Cⁿ⁺¹, we discuss the problem of classifying all local CR diffeomorphisms between open subsets N, N′ ⊂ M. Our method exploits the Tanaka-Webster pseudohermitian invariants of a contact form ϑ on M, their transformation formulae, and the Chern-Moser invariants. Our main application concerns a class of generalized ellipsoids where we classify all local CR mappings.

Let F be a C^∞-foliation on a compact C^∞-manifold M. We consider the group of all leaf preserving C^∞-diffeomorphisms of (M, F) which are isotopic to the identity through leaf preserving C^∞-diffeomorphisms. Then we show that the group is simple if and only if all leaves of F are dense.

This article is the third in a series of our investigation on a complete non-compact connected Riemannian manifold $M$. In the first series [KT1], we showed that all Busemann functions on an $M$ which is not less curved than a von Mangoldt surface of revolution $\widetilde{M}$ are exhaustions, if the total curvature of $\widetilde{M}$ is greater than π. A von Mangoldt surface of revolution is, by definition, a complete surface of revolution homeomorphic to $\boldsymbol{R}^{2}$ whose Gaussian curvature is non-increasing along each meridian. Our purpose of this series is to generalize the main theorem in [KT1] to an $M$ which is not less curved than a more general surface of revolution.

It is shown that there exists a Riemann surface on which every Dirichlet finite harmonic function is automatically bounded and yet the linear dimension of the linear space of Dirichlet finite harmonic functions on it is infinite.

A (holomorphic) involution σ of an Enriques surface S is said to be numerically reflective if it acts on the cohomology group H²(S, Q) as a reflection. We show that the invariant sublattice H(S, σ; Z) of the anti-Enriques lattice H^-(S, Z) under the action of σ is isomorphic to either 〈-4〉 ⊥ U(2) ⊥ U(2) or 〈-4〉 ⊥ U(2) ⊥ U. Moreover, when H(S, σ; Z) is isomorphic to 〈-4〉 ⊥ U(2) ⊥ U(2), we describe (S, σ) geometrically in terms of a curve of genus two and a Göpel subgroup of its Jacobian.

We derive explicit and simple conditions which in many cases allow one to decide, whether or not a Denjoy domain endowed with the Poincaré or quasihyperbolic metric is Gromov hyperbolic. The criteria are based on the Euclidean size of the complement. As a corollary, the main theorem allows us to deduce the non-hyperbolicity of any periodic Denjoy domain.

Let Ω be a Brelot space satisfying the domination principle and has a positive potential. We define the superharmonic extension property (SEP) for Ω and introduce a sufficient condition under which Ω has the SEP property. We give a characterization of the extreme Jensen measures in a Brelot space with SEP property.

Differential operators on Siegel modular forms which behave well under the restriction of the domain are essentially intertwining operators of the tensor product of holomorphic discrete series to its irreducible components. These are characterized by polynomials in the tensor of pluriharmonic polynomials with some invariance properties. We give a concrete study of such polynomials in the case of the restriction from Siegel upper half space of degree 2n to the product of degree n. These generalize the Gegenbauer polynomials which appear for n = 1. We also describe their radial parts parametrization and differential equations which they satisfy, and show that these differential equations give holonomic systems of rank 2ⁿ.

Let p be an odd prime number and ℓ a prime number with ℓ ≠ p. Let K_n = Q(ζ_pⁿ⁺¹) be the pⁿ⁺¹-st cyclotomic field. Let h_n and h_n^- be the class number and the relative class number of K_n, respectively. When ℓ = 2, we give an explicit bound m_p depending on p such that the ratio h_n^-/h_n-1^- is odd for all n > m_p. When ℓ ≥ 3, we also give a corresponding result on the ℓ-part of the relative class number of K_n⁺(ζ_ℓ). As an application, we show that when p ≤ 509, the ratio h_n/h₀ is odd for all n ≥ 1.