By Yuri Gliklikh

This e-book supplies a standard therapy to 3 parts of program of world research to Mathematical Physics formerly thought of particularly far-off from one another. those components are the geometry of manifolds utilized to classical mechanics, stochastic differential geometry utilized in quantum and statistical mechanics, and infinite-dimensional differential geometry basic for hydrodynamics.

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C. R. Acad. Sci. Paris Sr. , 331(9): 669–674 (2000). N. Monod: Continuous bounded cohomology of locally compact groups. Lecture Notes in Mathematics, 1758. Springer-Verlag, Berlin, 2001. D. Witte Morris: Can lattices in SL(n, R) act on the circle? This volume. A. Navas: Quelques nouveaux phénomènes de rang 1 pour les groupes de difféomorphismes du cercle. Comment. Math. , 80(2): 355–375 (2005). D. Witte, R. Zimmer: Actions of semisimple Lie groups on circle bundles. Geom. Dedicata, 87(1–3): 91–121 (2001).

The open dense orbit U identiﬁes with a homogeneous space G/H, where H is a closed subgroup of G. Consider X1 , X2 , . . , Xn global Killing ﬁelds on M that are linearly independent at some point of the open orbit U. As before, vol(X1 , X2 , . . , Xn ) is a nonzero constant, where vol is the holomorphic volume form associated to φ. Thus the Xi give a holomorphic parallelization of TM and Wang’s theorem enables us to conclude as in the previous proof. 3. [15] Let M be a compact connected simply connected complex nmanifold without nonconstant meromorphic functions and admitting a holomorphic rigid geometric structure φ.

It is minimal, unbounded, and strongly proximal. 13, and πϕ , πψ the corresponding homomorphisms. 13. Then, h πϕ (g) = πψ (g) h ∀g ∈ G. Proof. Assertion 2) is clear and follows from the various equivariance properties and thus we concentrate on 1). e. x ∈ B, the map g −→ πϕ (g)(ϕ(x)) = ϕ(gx) is measurable and hence the homomorphism πϕ : G → Homeo+ (S1 ) is measurable. Since G is locally compact second countable and since Homeo+ (S1 ) is second countable we deduce that πϕ is continuous. 1 we see that πϕ∗ (ebR ) = 0 and hence πϕ is nonelementary.