Showing papers in "Annals of Mathematics in 1970"

Groups of diffeomorphisms and the motion of an incompressible fluid

Abstract: In this paper we are concerned with the manifold structure of certain groups of diffeomorphisms, and with the use of this structure to obtain sharp existence and uniqueness theorems for the classical equations for an incompressible fluid (both viscous and non-viscous) on a compact C^∞ riemannian, oriented n-manifold M, possibly with boundary.

A model of set-theory in which every set of reals is Lebesgue measurable*

Abstract: We show that the existence of a non-Lebesgue measurable set cannot be proved in Zermelo-Frankel set theory (ZF) if use of the axiom of choice is disallowed. In fact, even adjoining an axiom DC to ZF, which allows countably many consecutive choices, does not create a theory strong enough to construct a non-measurable set. Let ZFC be Zermelo-Frankel set theory together with the axiom of choice. Let I be the statement: There is an inaccessible cardinal'.

Compounding Clifford's Theory

Abstract: Let G be a group, N be a normal subgroup of G, and p be an irreducible (finite-dimensional) character of N in some algebraically closed field a. Assume that the stabilizer G, of p in G has finite index in G. Then Clifford's theory [2] gives us a central extension G of the multiplicative group F of a by GIN together with a one-to-one correspondence between the set Ch (G I A) of all irreducible characters of G having p as an N-constituent and the set Ch (G ) of all projective irreducible characters of GjIN corresponding to the extension G , i.e., of all irreducible characters of G having the natural embedding of F in ! as F-constituents. Now we add another normal subgroup K of G containing N and an irreducible character T of K having p as an N-constituent. We may take K to be the inverse image in G of Ky/N. Then K is a normal subgroup of G . Let T e Ch (K I q) correspond to * e Ch (K ) under the Clifford correspondence. One easily verifies that Ch (G I A) c Ch (G I9) corresponds to Ch(G I A). Furthermore, the stabilizers G and G, are related by (0.la) G o is the inverse image in G of (G , G,)/N. (0. lb) G, = (G, n G,)K. It follows that [G : G +] is finite if and only if [G: G,] is. When this happens, we may apply Clifford's theory to G, K, and A , obtaining an extension G of F by G,/K and a one-to-one correspondence between Ch (G I A) and Ch (G ). We may also apply Clifford's theory to G , K , and *, obtaining an extension G of F by G ,/K and a one-to-one correspondence between Ch (G I A) and Ch (G ). By (0.1) there is a natural isomorphism of G ,/K (Gfl GW)/K? onto G,/K. So G and G are extensions of F by isomorphic groups. Furthermore both

On Some Games Which Are Relevant to the Theory of Recursively Enumerable Sets

Abstract: In ? 1 we shall describe a class of two-person infinite games called basic games which may be applied to the elementary theory T(9Z) of recursively enumerable (r.e.) sets. The theory T(ER) will be defined in ? 1 and has been studied previously by the author in [5] and [6]. A good source for background material is Rogers [9]. Our reason for studying basic games is that every theorem of T(ER) known at the present time can be proved by constructing an effective winning strategy for a suitable basic game. This contention will be supported in ? 2 by a number of examples. In ? 3 we discuss briefly the solution of a particular kind of basic game. In ? 4 we show that two natural subclasses of the class of basic games are not adequate for deriving all theorems of T(9I). In ? 5 we have summarized our reasons for thinking that the gametheoretic approach to recursion theory is a useful one, and have listed some open questions. I am grateful to the referee for many valuable suggestions regarding the format of this paper.

Curvature and Metric

Abstract: The theorema egregium or, in essence, the fundamental theorem of riemannian geometry asserts that curvature is an invariant of the metric. We ask the converse: how far does curvature determine the metric? Important theorems in this direction are the classical theorems for (embedded) surfaces. More recently there is a local theorem of E. Cartan and its global formulation due to W. Ambrose (cf. [1]). For a different approach see Nomizu and Yano [8]. In these theorems there are non-trivial hypotheses about the curvature tensor. We ask a more naive, but geometrically fascinating question: let (M, g), (M, U) be two Riemann manifolds. Denote the corresponding sectional curvatures by K respectively K. We say, M, M are isocurved if there exists a "sectional-curvature-preserving" diffeomorphism f: M O M, i.e., for every p e M and for every a, a 2-plane section of the tangent space Tp(M), we have K(a) = K(f* a) .

Period Relations of Schottky Type on Riemann Surfaces

Abstract: It was recognized in Riemann's work more than one hundred years ago and proved recently by Rauch, cf. [R2], that the g(g + 1)/2 unnormalized periods of the normal differentials of first kind on a compact Riemann surface S of genus g > 2 with respect to a canonical homology basis are holomorphic functions of 3g 3 complex variables, "the" moduli, which parametrize the space of Riemann surfaces near S and, hence, that there are (g 2)(g 3)/2 holomorphic relations among those periods. Eighty years ago, Schottky [S1] exhibited the relation for g = 4 as the vanishing of an explicit homogeneous polynomial in the Riemann theta constants. Sixty years ago, Schottky and Jung [SJ] conjectured a result which implies Schottky's earlier one and some generalizations for higher genera. Here, we formulate Schottky and Jung's conjecture precisely and, on the basis of a recent result of Farkas [F3], [F4], prove it. We then derive Schottky's result (we believe for the first time correctly) and exhibit a typical relation of this kind for g = 5 (we show how to do this for any genus). We do not prove that our relations imply all relations, but there are some indications that they do, indications to be dealt with in subsequent publications. The present paper is a consolidation and expansion of the notes [RF1] and [FR]. Our main result is formulated in ? 3 as Theorem 1 which asserts the proportionality of the squares of one set of theta constants, the Schottky constants, to certain two-term products of another set, the Riemann theta constants, both sets attached to S with a definite canonical homology basis and both defined in ? 2. Section 2 also contains other essential preliminary definitions and lemmata. In ? 5 we attain the principal object of the whole investigation by showing how, for g ? 4, the substitution of the proportionalities of Theorem 1 into suitable identities for general (g 1)-theta constants, in particular, for the Schottky theta constants leads immediately to relations among the Riemann

An Analogue of the Borel-Weil-Bott Theorem for Hermitian Symmetric Pairs of Non-Compact Type

Abstract: Let G be a connected non-compact semi-simple Lie group admitting a finite dimensional faithful representation. Let K be a maximal compact subgroup of G. We suppose that GIK is hermitian symmetric. When G -SL(2, R), V. Bargmann [31 constructed the discrete series for G on spaces of square-integrable holomorphic functions on GIK (which is isomorphic to the unit disc) and in the general case Harish-Chandra (cf. [5]) constructed part of the discrete series for G in a similar way. (See also [6].) However, in the general case, this method does not yield all the discrete series. In analogy with the Borel-Weil-Bott theorem [4, 8(a)], it was suggested in [91 that all the discrete classes (which were obtained by Harish-Chandra in [7(f)]) might be realized on spaces of square-integrable harmonic forms of type (0, q) ("square-integrable D-cohomology spaces") with coefficients in holomorphic vector bundles on GIK arising from finite dimensional irreducible unitary representations of K. (When q = 0, we have the case which was considered in [5].) We prove in this paper that most of the discrete classes are obtained in this way. We proceed to describe the results of this paper in more detail. Let g and f be the Lie algebras of G and K respectively. Let t be a Cartan subalgebra of t. Then t is also a Cartan subalgebra of g. Choose an ordering on the roots of (gC, tc) compatible with the complex structure on GIK. For an irreducible unitary representation z-A of K with the highest weight A, we denote by EA the holomorphic vector bundle on GIK associated to the contragredient representation. Let H,,q(EA) denote the Hilbert space of square-integrable harmonic forms of type (0, q) with coefficients in EA (these are the "squareintegrable a-cohomology spaces attached to EA" defined in [11]). The unitary representation wr of G on H,,q(EA) decomposes into a finite number of irreducible representations each of which belongs to the discrete series for G (Proposition 3.1). Let p denote the half sum of positive roots of (gC, tc) and let qA

Complements of Minimal Spanning Surfaces of Knots are Not Unique

Abstract: Every polyhedral simple closed curve in S3 bounds an orientable surface [2]; that is, if k is a knot, then there exists an orientable surface S such that D(S) = k. The minimal genus of all such spanning surfaces is the genus of the knot and a minimal spanning surface is a spanning surface with minimal genus. For the Neuwirth-Stallings [4, 5] knots, it is known that such a minimal surface is unique in the sense that there is a space homeomorphism throwing one minimal surface to any other. The important property here is that the commutator subgroup of the knot group is finitely generated and, hence, free [4]. Stallings then shows that S3 k fibers over S' with fiber a minimal spanning surface of k. Such a fibration can then be used to show the uniqueness of the minimal spanning surface. Hale Trotter, in a letter to C. D. Feustel, indicates that there is an example of a knot with two different minimal surfaces. His example depends heavily on the existence of non-invertible knots [6]. Trotter's example has the property that S3 S1 is homeomorphic to S3 S2. The purpose of this note is to construct a knot k with two minimal spanning surfaces, SI and S2, such that S3 Si are not homeomorphic. The proof actually shows that Z1(S3 Si) are not isomorphic. The author is indebted to Professor C. B. Schaufele, Dr. C. D. Feustel, and Mr. A. C. Connor for helpful conversations.