Katsumi Nomizu

The existence of complete Riemannian metrics

The purpose of the present note is to prove the following results. Let M be a connected differentiable manifold which satisfies the second axiom of countability. Then (i) M admits a complete Riemannian metric; (ii) If every Riemannian metric on M is complete, M must be compact. In fact, somewhat stronger results will be given as Theorems 1 and 2 below. Let M be a connected differentiable manifold. It is known that if M satisfies the second axiom of countability, then M admits a Riemannian metric. Conversely, it can be shown that the existence of a Riemannian metric on M implies that M satisfies the countability axiom. For any Riemannian metric g on M, we can define a natural metric d on M by setting the distance d(x, y) between two points x and y to be the infinimum of the lengths of all piecewise differentiable curves joining x and y. The Riemannian metric g is complete if the metric space M with d is complete. It is known that this is the case if and only if every bounded subset of M (with respect to d) is relatively compact. We shall say that a Riemannian metric g is bounded if M is bounded with respect to the metric d. We shall prove

Trigonometry in Lorentzian Geometry

(1984). Trigonometry in Lorentzian Geometry. The American Mathematical Monthly: Vol. 91, No. 9, pp. 543-549.

Conjugate Connections and Radon's Theorem in Affine Differential Geometry.

For a given nondegenerate hypersurfaceMn in affine space ℝn+1 there exist an affine connection ∇, called the induced connection, and a nondegenerate metrich, called the affine metric, which are uniquely determined. The cubic formC=∇h is totally symmetric and satisfies the so-called apolarity condition relative toh. A natural question is, conversely, given an affine connection ∇ and a nondegenerate metrich on a differentiable manifoldMn such that ∇h is totally symmetric and satisfies the apolarity condition relative toh, canMn be locally immersed in ℝn+1 in such a way that (∇,h) is realized as the induced structure?In 1918J. Radon gave a necessary and sufficient condition (somewhat complicated) for the problem in the casen=2. The purpose of the present paper is to give a necessary and sufficient condition for the problem in casesn=2 andn≥3 in terms of the curvature tensorR of the connection ∇. We also provide another formulation valid for all dimensionsn: A necessary and sufficient condition for the realizability of (∇,h) is that the conjugate connection of ∇ relative toh is projectively flat.

Some results in E. Cartan’s theory of isoparametric families of hypersurfaces

The purpose of this department is to provide early announcement of significant new results, with some indications of proof. Although ordinarily a research announcement should be a brief summary of a paper to be published in full elsewhere, papers giving complete proofs of results of exceptional interest are also solicited. Manuscripts more than eight typewritten double spaced pages long will not be considered as acceptable. All research announcements are communicated by members of the Council of the American Mathematical Society. An author should send his paper directly to a Council member for consideration as a research announcement. A list of members of the Council for 1974 is given at the end of this issue.

Characteristic roots and vectors of a diifferentiable family of symmetric matrices

LetA(x) be a differentiable family of k × k symmetric matrices where x runs through a domain D in R nWe prove that if λ is a continuous function onDsuch that, for every x ϵD,λ(x) is a characteristic root of A(x) of constant multiplicity m, then λ is a differentiable function and there exists, locally, a differentiable family of ortho-normal bases for the eigenspace. The case n = 1 has been known in the standard treatises on the perturbation theory for linear operators.

A NEW EQUIAFFINE THEORY FOR SURFACES IN ℝ4

In this paper, we investigate the geometry of nondegenerate affine surfaces in ℝ4. The main idea is to introduce an equiaffine structure on the surface by constructing a canonical transversal plane field with the aid of the affine metric. As an application, we then investigate surfaces with vanishing cubic forms. If the affine metric is positive-definite, such a surface can be locally described as a complex curve in . On the other hand, if the affine metric is indefinite, such a surface can be seen as the product of two planar curves.

Centroaffine immersions of codimension two and projective hypersurface theory

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 Ḿ with an affine connection ∇ is called an affine immersion if there is a transversal vector field ξ such that ∇ x f * (Y) = f * (∇ x Y) + h(X,Y)ξ holds for any vector fields X, Y on M n . When f : M n → R n+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.

On Flat Surfaces in S 1 3 and H 1 3

The main purpose of the present paper is to study isometric immersions of the Euclidean plane E2 and the Lorentzian plane L2 into the 3-dimensional Lorentzian manifolds S3 1 and H3 1, of constant sectional curvature 1 and -1, respectively.

Un théorème sur les groupes d'holonomie

MM. Ozeki et Hano ont recemment demontre [3] que tout sous-groupe connexe de Lie du groupe lineaire GL(n, R), n≧2 , peut etre realise comme groupe d’holonomie d’une certaine connexion lineaire dans un espace affine de dimension n .

Another Rigidity Theorem for Affine Immersions

The theorem of Beez-Killing in Euclidean differential geometry states as follows [KN, p.46]. Let f: Mn → Rn+1 be an isometric immersion of an n-dimensional Riemannian manifold into a Euclidean (n + l)-space. If the rank of the second fundamental form of f is greater than 2 at every point, then any isometric immersion of Mn into Rn + 1 is congruent to f. A generalization of this classical theorem to affine differential geometry has been given in [O] (see Theorem 1.5). We shall give in this paper another version of rigidity theorem for affine immersions.