This paper has been motivated by various problems and results in differential geometry. The main motivation is the study of curvature homogeneous Riemannian spaces initiated in 1960 by I.M. Singer (see Section 9—Appendix for the precise definitions and references). Up to recently, only sporadic classes of examples have been known of curvature homogeneous spaces which are not locally homogeneous. For instance, isoparametric hypersurfaces in space forms give nice examples of nontrivial curvature homogeneous spaces (see [FKM]). To study the topography of curvature homogeneous spaces more systematically, it is natural to start with the dimension n = 3. The following results and problems have been particularly inspiring.

In this note we give the structure of maximally differential ideals in a Noetherian local ring of prime characteristic p > 0, in terms of their generators. More precisely, we prove the following result:

THEOREM 4. Let A be a Noetherian local ring of prime characteristic p > 0 with maximal ideal m. Let I be a proper ideal of A. Suppose n= emdim(A) and r = emdim(A/l). If I is maximally differential under a set of derivations of A then there exists a minimal set xl,…,xn of generators of m such that I = (xρl, …,xρr, xr+1,…xn).

Consider the function with variables z1…,zm

where υ and λj (j = 1,…,m) are complex numbers and.Г is a suitably chosen integral domain. In case m = 2, if we set [z1 z2]n as Г, it is the Selberg integral [22]. Our function can be regarded as an extention of it; so we may call (0.1) a Selberg type integral It is known that (0.1) satisfies a Gauss-Manin system, i.e. a system of rationally holonomic differential equations [3], [21].

Affine differential geometry developed by Blaschke and his school [B] has been reorganized in the last several years as geometry of affine immersions. An immersion f of an n-dimensional manifold M with an affine connection ∇ into an (n + 1)-dimensional manifold Ḿwith an affine connection ∇ is called an affine immersion if there is a transversal vector field ξ such that ∇xf*(Y) = f*(∇xY) + h(X,Y)ξ holds for any vector fields X, Y on Mn. When f: Mn→ Rn+1 is a nondegenerate hypersurface, there is a uniquely determined transversal vector field ξ, called the affine normal field, an essential starting point in classical affine differential geometry. The new point of view allows us to relax the non-degeneracy condition and gives us more freedom in choosing ξ; what this new viewpoint can accomplish in relating affine differential geometry to Riemannian geometry and projective differential geometry can be seen from [NP1], [NP2], [NS] and others. For the definitions and basic formulas on affine immersions, centroaffine immersions, conormal (or dual) maps, projective flatness, etc., the reader is referred to [NP1]. These notions will be generalized to codimension 2 in this paper.

Let ρ be an algebraic homomorphism of a linear algebraic group G into the affine transformation group Aff(V) of a finite dimensional vector space V. We say that a triplet (G, V, ρ) is a prehomogeneous affine space, if there exists a proper algebraic subset S of V such that V — S is a single ρ(G)-orbit. In particular, (G, V, ρ) is a usual prehomogeneous vector space (PV, briefly) in the case where ρ(G) ⊂ GL(V) (cf. [5], [7]). In the preceding paper [2], we defined zeta functions associated with certain prehomogeneous affine spaces and proved their analytic continuation and functional equations.

In [3] Mori gives a description of all extremal rays (extremal morphisms) arising on a smooth projective threefold with a numerically non-effective canonical bundle. Generally speaking, every smooth projective threefold V with a numerically non-effective canonical class Kv admits an extremal morphism π: V— → Y. The assumption that V admits a non-trivial pair of extremal morph-isms

imposes strong conditions on V. This is the essence of the Theorem 1.5 of the present work. In particular, we obtain a description of the threefolds which admit two biregular structures of conic bundles over non-singular surfaces S1 = Y1 and S2 = Y2. By the results of §3 the surfaces Sl and S2 must be either ruled surfaces with isomorphic basic curves, or S1 ≈ S2 ≈ P2.

Consider an end Ω in the sense of Heins (cf. Heins [3]): Ω is a relatively non-compact subregion of an open Riemann surface such that the relative boundary ∂Ω consists of finitely many analytic Jordan closed curves, there exist no non-constant bounded harmonic functions with vanishing boundary values on ∂Ω and Ω has a single ideal boundary component. A density P = P(z)dxdy (z = x + iy) is a 2-form on Ω∩∂Ω with nonnegative locally Holder continuous coefficient P(z).

In this paper it is shown that there is a unimodal Levy process with oscillating mode. After the author first constructed an example of such a self-decomposable process, Sato pointed out that it belongs to the class of semi-stable processes with β < 0. We prove that all non-symmetric semi-stable self-decomposable processes with β < 0 have oscillating modes.

Let (M, g) be a Riemannian manifold of dimension n≥ 3 and ĝanother metric on M which is pointwise conformai to g. It can be written where u is a positive smooth function on M. Then the curvature of g is computable in terms of that of g and the derivatives of u up to second order. In particular, if S and S denote the scalar curvature of g and g respectively, they are related by the equation

where ▽u denotes the Laplacian of u, defined with respect to the metric g.

In recent papers [10, 11, 12, 13, 14], we defined some new ρ-invariants for any rational prime ρ congruent to 1 mod 4 and D-invariants for any positive square-free integer D such that the fundamental unit εD of real quadratic field Q(√D) satisfies NεD = –1, and studied relationships among these new invariants and already known invariants.

One of our main purposes in this paper is to generalize these D-invariants to invariants valid for all square-free positive integers containing D with NεD = 1. Another is to provide an improvement of the theorem in [14] related closely to class number one problem of real quadratic fields. Namely, we provide, in a sense, a most appreciable estimation of the fundamental unit to be able to apply, as usual (cf. [3, 4, 5, 9, 12, 13]), Tatuzawa’s lower bound of L(l, XD) (Cf[7]) for estimating the class number of Q(√D) from below by using Dirichlet’s classical class number formula.