By Yuri Gliklikh

ISBN-10: 0387948678

ISBN-13: 9780387948676

The first variation of this e-book entitled research on Riemannian Manifolds and a few difficulties of Mathematical Physics was once released through Voronezh Univer sity Press in 1989. For its English version, the publication has been considerably revised and multiplied. particularly, new fabric has been further to Sections 19 and 20. i'm thankful to Viktor L. Ginzburg for his labor at the transla tion and for writing Appendix F, and to Tomasz Zastawniak for his a variety of feedback. My unique thank you visit the referee for his worthwhile comments at the concept of stochastic strategies. ultimately, i want to recognize the help of the AMS fSU relief Fund and the overseas technological know-how origin (Grant NZBOOO), which made attainable my paintings on the various new effects integrated within the English version of the e-book. Voronezh, Russia Yuri Gliklikh September, 1995 Preface to the Russian variation the current ebook is outwardly the 1st in monographic literature during which a standard therapy is given to 3 parts of worldwide research formerly consid ered fairly far away from one another, specifically, differential geometry and classical mechanics, stochastic differential geometry and statistical and quantum me chanics, and infinite-dimensional differential geometry of teams of diffeomor phisms and hydrodynamics. The unification of those subject matters below the canopy of 1 ebook appears to be like, even if, rather average, because the exposition is predicated on a geometrically invariant type of the Newton equation and its analogs taken as a primary legislations of motion.

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**Example text**

The reduced covariant derivative V has all four properties of the regular covariant derivative. ) Proof. Since the operator P is linear on fibers of TM, only the fourth deserves a proof. For admissible X, Y and a smooth function f, we have Ox(fY) = PVx(fY) = P(fVxY+(Xf)Y) = fVxY+(Xf)Y , because PY = Y. Thus, V is the covariant derivative on admissible vectors and, in particular, it gives rise to the parallel translation of admissible vectors along admissible curves. The definition of such a parallel translation is quite similar to the standard one.

The fiber of H over a point (m, b) E OO(M) is formed by "infinitesimal" parallel translations of the frame e. It is easy to check that the subbundle is invariant with respect to the right action of O(k), and the fibers of H have zero intersection with the vertical subspaces V(m,b). Thus, H can be thought of as an analog of a connection. 24 Chapter 2. 4. The subbundle H is called a reduced connection. 2. If the constraint is holonomic, the reduced connection is, in fact, the Levi-Civita connection on the integral manifolds with respect to the induced Riemannian metric.

The set of all possible positions of the rigid body with a stationary point, is the special orthogonal group SO(3). Fixing an orthonormal basis in IR3, we may identify SO(3) with the group of all orthogonal matrices with unit determinant. , the tangent space to SO(3) at e = id) is formed by skew-symmetric 3 x 3 matrices. Thus so(3) can be identified with W. 3, respectively. The Riemannian metric obtained from the Killing form by left translations turns out to be bi-invariant. This metric is used as the canonical one to express the kinetic energy via the inertia tensor.

### Global Analysis in Mathematical Physics: Geometric and Stochastic Methods by Yuri Gliklikh

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