Differential Geometry and Relativity: A Volume in Honour of by R. Couty, A. Revuz (auth.), M. Cahen, M. Flato (eds.)

By R. Couty, A. Revuz (auth.), M. Cahen, M. Flato (eds.)

On the celebration of the 60th birthday of Andre Lichnerowicz a few his associates, a lot of whom were his scholars or coworkers, determined to rejoice this occasion by means of getting ready a jubilee quantity of contributed articles within the major fields of study marked by means of Lichnerowicz's paintings, particularly differential geometry and mathematical physics. boundaries of house and time didn't permit us to incorporate papers from all Lichnerowicz's acquaintances nor from all his former scholars. It was once both very unlikely to mirror in one e-book the nice number of matters tackled by means of Lichnerowicz. regardless of those boundaries, we are hoping that this booklet displays many of the current traits of fields during which he labored, and a few of the themes to which he contributed in his lengthy - and never but entire - occupation. This occupation was once a great deal marked via the effect of his masters, Elie Cartan who brought him to investigate in arithmetic, often in geometry and its family with mathematical physics, and Georges Darmois who constructed his curiosity for mechanics and physics, particularly the speculation of relativity and electromagnetism. This par­ ticular mixture, and his own expertise, made up of him a traditional clinical inheritor and continuator of the French mathematical physics tuition within the culture of Henri Poincare. a few of his works could also be top certified through a brand new box identify, that of actual ma­ thematics: branches of natural arithmetic totally influenced through physics.

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6. REMARK. One always has i(g) ~ 2c(g). We prove it by contradiction: suppose ;(g) < 2c(g). Let m, n be points with d(m, n) = i(g), let y be a segment from m to n and denote by p the middle point of y; by assumption m,n E B(p, c(g» which is a convex ball. 5 implies that m, n are conjugate on y. By a classical result (see [1], pp. 237-238), if one parametrizes y with y(O) = m, y(t) = nand extends it beyond t for every t' > t, then there exists a segment u t ' from m to y(t') = n' of length d(m, n') < t' and, moreover, the uniqueness of segments from m to n implies that U t ' converges to y when t' tends to t; in particular we can choose U t ' as close to the nonsegment geodesic yllO,t'j as we want.

O»}, where and JOSEPH A. WOLF «» indicates that the term occurs just when 1= 2v + 1. Now the representation of K on the tangent space of M' is Tt(8)T2, which has weight system {±(aj + bd, ± (aj - bd, «± aj»}, 1 ~ j ~ u and 1 ~ k ~ v. That has Pfaffian and E (,0') is the Euler class on M'. 4 to the case 1 = 2v. The fact that c is integral and K"-invariant, says that, on the Lie algebra of the maximal torus, c is an integral polynomial in the elementary symmetric functions lTs(ai, ... , a;) and the IT,(bi, ...

B. , 'Compact Manifolds of Nonpositive Curvature', J. DifJ. Geometry 7 (1972), 211-228. [6] Ramanathan, K. , 'Discontinuous Groups II', Nachr. Akad. Wiss. Gottingen, II. Math-Phys. Klasse, (1964), 145-164. [7] Wang, H. , 'On a Maximality Property of Discrete Subgroups with Fundamental Domain of Finite Measure, Amer. J. Math. 89 (1967), 124-132. [8] Yau, S. , 'Non-Existence of Continuous Convex Functions on Certain Riemannian Manifolds', Math. Ann. 207 (1974), 269-270. AVEZ HARMONIC FUNCTIONS ON GROUPS* ABSTRACT.

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