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

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

On the social gathering of the 60th birthday of Andre Lichnerowicz a few his buddies, a lot of whom were his scholars or coworkers, made up our minds to have a good time this occasion by means of getting ready a jubilee quantity of contributed articles within the major fields of analysis marked via 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 associates nor from all his former scholars. It was once both very unlikely to mirror in one publication the nice number of matters tackled by way of Lichnerowicz. despite those obstacles, we are hoping that this booklet displays a number of the current traits of fields within which he labored, and a few of the themes to which he contributed in his lengthy - and never but comprehensive - profession. This profession used to be greatly marked via the impression of his masters, Elie Cartan who brought him to analyze in arithmetic, commonly in geometry and its family with mathematical physics, and Georges Darmois who built his curiosity for mechanics and physics, particularly the idea of relativity and electromagnetism. This par­ ticular blend, and his own expertise, made up of him a usual clinical inheritor and continuator of the French mathematical physics college within the culture of Henri Poincare. a few of his works might also be most sensible certified via a brand new box identify, that of actual ma­ thematics: branches of natural arithmetic completely inspired by means of physics.

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Note that the Killing vector fields in Lemma 2 generate a torus group since I(M) is compact by Lemma 3, and this fact is consistent with the structure theorem in [5]. In connection with the conjecture stated in § 1, we can ask if the following complex analogue of Lemma 1 is true. If M is a complete Kahler manifold with finite volume, are all plurisubharmonic functions on M constant? If this is true, we obtain the complex analogue of Lemma 2 to the effect that on a complete Kahler manifold of nonpositive holomorphic bisectional curvature with finite volume every holomorphic vector field is parallel.

Croissance polynomiale et periodes des fonctions harmoniques', Bull. Soc. Math. France 101 (1973), 333-379. , 'A Note on Curvature and Fundamental Group', I. DiJf. Geometry 2 (1968), 1-7. , 'A Proof of Liouville's Theorem', Proc. S. 12 (1969), 995. [9] Svarc, A. , 'A Volume Invariant of Coverings', Dokl. Akad. Nauk. R. 105 (1955), 32-34. , 'Growth of Finitely Generated Groups and Curvature of Riemannian Manifolds', I. DiJf. Geom. 2 (1968), 421-446. [11] Yau, S. , Harmonic Functions on Complete Riemannian Manifolds', Comm.

T) ~(4'1Tt)-d/2(a~p... t + ... t .. +O(t .. 4b) Our main result is the observation. 5. THEOREM. " = (- 1) k Vol (M) , Vol (M') ak....... Proof. " is a local invariant of M, that is ak. ,,(x) is a polynomial in the curvatures of M and E at x and their covariant derivatives. ,p(x) is homogeneous of degree k in the curvatures. ,,(x) . Vol(M). If y E M', the same considerations hold, and the curvatures are the negatives of those at x EM, so a ~..... Ax)· Vol (M'). D. ,p is an algebraic problem with the Peter-Weyl Theorem for G', Cartan's highest weight theory for the representations of G', Frobenius' Reciprocity Theorem, and the decomposition of a representation of G' under restriction to K.

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