In this paper we have given a biholomorphic classification of smooth, connected, protective, non-ruled surfaces X with a smooth, connected, hyperplane section C relative to L, where L is a very ample line bundle on X, such that g = g(C) = g(L) is less than or equal to six. For a similar classification of rational surfaces with the same conditions see [Li].

One of the basic questions of Riemannian geometry is that “If two Riemannian manifolds are similar with respect to the Riemannian invariants, for example, the curvature, the volume, the first eigenvalue of the Laplacian, then are they topologically similar?”. Initiated by H. Rauch, many works are developed to the above question. Recently M. Gromov showed a remarkable theorem ([7] 8.25, 8.28), which may be useful not only for the above question but also beyond the above. But it seems to the author that his proof is heuristic and it contains some gaps (for these, see § 1), so we give a detailed proof of 8.25 in [7]. This is the first purpose of this paper. Second purpose is to prove a differentiable sphere theorem for manifolds of positive Ricci curvature, using the above theorem as a main tool.

Let X and Y be any pure dimensional subschemes of Pnk over an algebraically closed field K and let I(X) and I(Y) be the largest homogeneous ideals in K[x0,…, xn] defining X and Y, respectively. By a pure dimensional subscheme X of Pnk we shall always mean a closed pure dimensional subscheme without imbedded components, i.e., all primes belonging to I(X) have the same dimension.

A method to show the existence of infinitely many closed geodesies on a manifold with non degenerate Riemannian metric is to use socalled divisibility lemma which is conjectured to hold in [K].

Our purpose in this series of papers is, then, to show first the divisibility lemma in a modified form on k-sphere Sk (k≧3) with (strongly) non degenerate Riemannian metric and to deduce the existence of infinitely many closed geodesies on Sk (k≧3) using the modified divisibility lemma by equivariant modifications of flows.

In connection with the Shimura correspondence, Shintani [6] and Niwa [4] constructed a modular form by the integral with the theta kernel arising from the Weil representation. They treated the group Sp(1) × O(2, 1). Using the special isomorphism of O(2, 1) onto SL(2), Shintani constructed a modular form of half-integral weight from that of integral weight. We can write symbolically his case as “O(2, 1)→ Sp(1)” Then Niwa’s case is “Sp(l)→ O(2, 1)”, that is from the halfintegral to the integral. Their methods are generalized by many authors. In particular, Niwa’s are fully extended by Rallis-Schiffmann to “Sp(l)→O(p, q)”.

Let R⊂ R′ be an “unramified” extension of discrete valuation rings in the sense that a local parameter p of R is also a local parameter in R⊂. Suppose that the inclusion R→R⊂ induces separable extensions on fraction and residue fields.

It is a well-known result due to Sjogren [9] that if G is a finitely generated p-group then, for all n ≦ − 1, the (n + 2)-th dimension sub-group Dn+2(G) of G coincides with γn+2(G), the (n + 2)-th term of the lower central series of G. This was earlier proved by Moran [5] for n ≦ p − 2. For p = 2, Sjogren’s result is the best possible as Rips [8] has exhibited a finite 2-group G for which D4G) ≠ γ4(G) (see also Tahara [10, 11]). In this note we prove that if G is a finitely generated metabelian p-group then, for all . It follows, in particular, that, for p odd, for all n ≦ p and all metabelian p-groups G.

Let ϕ: M → RN be an isometric imbedding of a compact, connected surface M into a Euclidean space RN. ψ is said to be umbilical at a point p of M if all principal curvatures are equal for any normal direction. It is known that if the Euler characteristic of M is not zero and N = 3, then ψ is umbilical at some point on M. In this paper we study umbilical points of surfaces of higher codimension. In Theorem 1, we show that if M is homeomorphic to either a 2-sphere or a 2-dimensional projective space and if the normal connection of ψ is flat, then ψ is umbilical at some point on M. In Section 2, we consider a surface M whose Gaussian curvature is positive constant. If the surface is compact and N = 3, Liebmann’s theorem says that it must be a round sphere. However, if N ≥ 4, the surface is not rigid: For any isometric imbedding Φ of R3 into R4 Φ(S2(r)) is a compact surface of constant positive Gaussian curvature 1/r2. We use Theorem 1 to show that if the normal connection of ψ is flat and the length of the mean curvature vector of ψ is constant, then ψ(M) is a round sphere in some R3 ⊂ RN. When N = 4, our conditions on ψ is satisfied if the mean curvature vector is parallel with respect to the normal connection. Our theorem fails if the surface is not compact, while the corresponding theorem holds locally for a surface with parallel mean curvature vector (See Remark (i) in Section 3).

This is a continuation of the previous paper [8] concerning the relation between the arithmetic of imaginary quadratic fields and cusp forms of weight one on a certain congruence subgroup. Let K be an imaginary quadratic field, say K = with a prime number q ≡ − 1 mod 8, and let h be the class number of K. By the classical theory of complex multiplication, the Hubert class field L of K can be generated by any one of the class invariants over K, which is necessarily an algebraic integer, and a defining equation of which is denoted by

Φ(x) = 0.

It is well known that the concept of energy has played a fruitful role in potential theory and Markov processes. Cartan’s work [6] led to kernel-free potential theories of Beurling-Deny [2]. Since then many authors have worked on this, M. Fukushima [8], M. Silverstein [16], J. Bliedner [3], Berg-Forst [1], to name some. In these works, however, the main thrust is Hubert space theoretic.