In [1] S. Kobayashi showed that the connected components of the set of zeros of a Killing vector field on a Riemannian manifold (Mn,g) are totally geodesic submanifolds of (Mn,g) of even codimension including the case of isolated singular points. The purpose of this short note is to give a simple proof of the corresponding result for conformal vector fields on compact Riemannian manifolds. In particular we prove the following

In a recent paper, [6], Nomizu and Rodriguez found a geometric characterization of umbilical submanifolds Mn ⊂ Rn+p in terms of the critical point behavior of a certain class of functions Lp, p ⊂ Rn+p , on Mn . In that case, if p ⊂ Rn+p, x ⊂ Mn , then Lp(x) = (d(x,p)) 2, where d is the Euclidean distance function.

The Lie algebra gr of all infinitesimal automorphisms of a Siegel domain in terms of polynomial vector fields was investigated by Kaup, Matsushima and Ochiai [6]. It was proved in [6] that gr is a graded Lie algebra; gr = g-1 + g-1/2 + g0 + g1/2 + g1 and the Lie subalgebra ga of all infinitesimal affine automorphisms is given by the graded subalgebra; ga = g-1 + g-1/2 + g 0. Nakajima [9] proved without the assumption of homogeneity that the non-affine parts g1/2 and g1 can be determined from the affine part ga .

Let V be a compact nonsingular algebraic variety of dimension n with a Hodge structure ω and let Hi,i(V,C) be the subgroup of 2i-th cohomology group H2i(V,C) represented by harmonic (i,i)-forms on V with respect to ω.

It is known (Pursell and Shanks [9]) that an isomorphism between Lie algebras of infinitesimal automorphisms of C∞ structures with compact support on manifolds M and M′ yields an isomorphism between C∞ structures of M and M’.

Omori [5] proved that this is still true for some other structures on manifolds. More precisely, let M and M′ be Hausdorff and finite dimensional manifolds without boundary. Let α be one of the following structures:

Soit X un groupe abélien localement compact et dénombrable à l’infini; ζ sera sa mesure de Haar. Dans les articles précédents [10] et [11], pour un noyau de convolution de Hunt N sur X, nous avons défini la famille sous-ordonnée H(N; X) au noyau N, qui est une large classe de noyaux de convolution de Hunt sur X définie par N et la totalité des noyaux de convolution de Hunt bornés sur la droite réelle R portés par R+ = {t ∈ R; t ≧ 0}.

As is well known, the Hopf fibration S 3 → S 2 is the restriction of the map h: R 4 → R 3 given by , a system of three quadratic forms [2]. Since spheres and the map are defined by polynomials with coefficients in Q, the original setting can be considered as a localization at infinity of the underlying algebraic sets and morphism defined over Q. This makes one think of the arithmetic of the Hopf maps.

In der vorliegenden Arbeit konstruieren wir, indem wir die Schlüsse von Armitage [1] in einer für unseren Zweck geeigneten Form erbringen, eine andere Klasse von L-Funktionen als bei Armitage [1] und Fröhlich [5], welche an der Stelle s = 1/2 eine Nullstelle ungerader Ordnung haben.

The manifold in this paper is assumed to be connected differentiable of class C∞. Let Dr(M) and Ӿr(M) be the set of all diffeomorphisms and vector fields of class Cr on a manifold M with Whitney Cr topology, respectively. In [2], the concept of weak stability is defined. The definition is equivalent to the following ((2.1) of this paper); f∈Dr(M) or X ∈ Ӿr(M) is weakly (allowably) stable if and only if there is a neighborhood U of f or X in Dr(M) or Ӿr(M) such that for any (a suitable) g or Y ∈ U the set of all elements topologically equivalent to g or Y is dense in U, respectively. Here, f, g ∈ Dr(M) are said to be topologically equivalent if they are topologically conjugate and X, Y ∈ Ӿr(M) are said to be topologically equivalent if there is a homeomorphism mapping any trajectory of X onto a trajectory of Y preserving the orientations of the trajectories. Similarly, weak Ω-stability is defined for f and X.

We shall prove in Chapter I the hypoellipticity for a class of degenerate elliptic operators of higher order. Chapter II will be devoted to the consideration of the regularity at the boundary for the solutions of general boundary problems for the equations considered in Chapter I being restricted to the second order.