The aim of this paper is to investigate the behaviour of Green lines at Royden’s boundary Γ of a Riemann surface R with the Green function g(z, o) with the fixed pole o in R. We denote by the totality of Green lines L issuing from the fixed point o.

The aim of this paper is to study a property of a special kind of exact functors and give some applications to projective modules and injective modules.

In section 1 we introduce the notion of faithfully exact functors [Definition 1] as a generalization of the functor T(X) = X⊗M, where M is a faithfully flat module, and give a property of this class of functors [Theorem 1.1]. Next, applying this general theory to functors ⊗ and Horn, we define the notion of faithfully projective modules [Definition 2] and faithfully injective modules [Definition 3]. In the commutative case “faithfully projective” means, however, simply “projective and faithfully flat” [Proposition 2.3]. In section 2, equivalent conditions for a projective module P to be faithfully projective are given [Theorem 2.2, Proposition 2. 3 and 2.4]. And a simpler proof is given to Y. Hinohara’s result [6] asserting that projective modules over an indecomposable weakly noetherian ring are faithfully flat [Proposition 2.5]. In section 3, we consider faithfully injective modules.

In the theory of automorphic functions for a properly discontinuous group G of linear transformations, the Poincaré theta series plays an essential role, since the convergence problem of the series occupies an important part of the theory. This problem was treated by many mathematicians such as Poincaré, Burnside [2], Fricke [4], Myrberg [6], [7] and others. Poincaré proved that the (-2m)-dimensional Poincaré theta series always converges if m is a positive integer greater than 2, and Burnside treated the problem and conjectured that ( -2)-dimensional Poincaré theta series always converges if G is a Schottky group. This conjecture was solved negatively by Myrberg. As is shown later (Theorem A), the convergence of Poincaré theta series gives an information on a metrical property of the singular set of the group.

Let M be an n-dimensional (n≥2) connected C∞-manifold with a linear connection. For simplicity, tensor fields on M will simply be called tensors on M. A tensor S on M is said to be parallel if its covariant derivative is everywhere zero in M, i.e., if ▽S = 0. S is said to be recurrent if its covariant derivative is equal to the tensor product of a covector and S itself, i.e., if ▽S = W⊗S, where W is called the recurrence covector.

Let Γ be a continuous pseudo group acting on a manifold M. Denote by (M, M’, ρ) a fibered manifold preserved by transformations in Γ. Then any f in Γ locally induces a local transformation f′ of M′. Denote by Γ/ρ the set of all such f′. Then it might seem natural to expect that Γ/ρ is a continuous pseudo group acting on M′. However the matter is not so simple. For instance, take f′ and g′ in Γ/ρ such that the composition f′°g′ can be defined. Then they can be lifted to local transformations f and g belonging to Γ.

It is a well-known result in the theory of conformal mappings of Jordan domains that if a domain D in the z-plane bounded by a closed Jordan curve C is mapped conformally on the disc |w;|<1 by a function w = f(z), analytic and univalent in D, then f(z) will be continuous on the closure of D and will map C on |w| =1 in a one to one manner (Carathéodory [2]), and that if C is rectifiable, then f(z) will map sets E of points of linear measure zero on C onto sets of linear measure zero on the circumference |w| = 1 and sets E of positive linear measure onto sets of positive linear measure on |w| =1 (F. and M. Riesz [12] and Lusin and Privaloff [8]).

Recently the important contributions to the cluster sets theory of the meromorphic functions in the unit-disc have been done by many authors. For its recent development, we refer to K. Noshiro [10]. Roughly speaking, these studies can be divided into two classes ; the first one is topological, and the second one is metrical. As far as the author knows, there exist very few results on the metrical theorems on cluster sets of functions meromophic in an arbitrary connected domain, except for the case that its boundary is of logarithmic capacity zero. (K. Noshiro [10] pp. 5-31).

Let D be a subregion of a Riemann surface F, whose relative boundary consists of at most countable number of analytic curves which do not cluster in F. For a regular exhaustion {Fn} of F, we put Dn = D∩ (F— Fn), and define the extremal radius R(P, ∂Dn) of the relative boundary ∂Dn of Dn, measured at a point P(∈F0) of F with respect to the connected component of F - Dn which contains P. Let K(|z|≦r) be a disk centered at P and contained in a parametric disk of P.

Recently J. L. Doob [2] evaluated the Dirichlet integral of the BLD harmonic funtion on a Green space in terms of its fine boundary values and θ-kernel of L. Naïm.

On the other hand, the general theory of additive functionals of Markov processes enables us to define the concept of the Dirichlet norm of functions with respect to Markov processes.

A time reversion of a Markov process was discussed by Kolmogoroff for Markov chains in 1936 [6] and for a diffusion in 1937 [7l He described it as a process having an adjoint transition probability. Although his treatment is purely analytical, in his case if the process xt has an invariant distribution, the reversed process zt = x-t is the process with the adjoint transition probability. In this discussion, however, it is very restrictive that the initial distribution of the process must be an invariant measure.

Let R be a hyperbolic Riemann surface and gw(z) be the Green function on R with its pole w in R. We denote by (R) the totality of sequences of points in R not accumulating in R and

When L is a parabolic differential operator of second order, Nirenberg [6] proved the maximum principle for the function u which has second order continuous derivatives and satisfies Lu≧0. Recently Friedman [2] has proved the maximum principle for the measurable function satisfying Lu≧O in the wide sense. This function is named a weakly L-subparabolic function. On the other hand, Littman [5] earlier than Friedman, has defined a weakly A- subharmonic function for an elliptic differential operator A of second order and has showed the maximum principle for it.