By Dmitri Burago, Yuri Burago, Sergei Ivanov
"Metric geometry" is an method of geometry in response to the suggestion of size on a topological house. This technique skilled a truly quickly improvement within the previous few a long time and penetrated into many different mathematical disciplines, reminiscent of workforce conception, dynamical platforms, and partial differential equations. the target of this graduate textbook is twofold: to provide a close exposition of uncomplicated notions and strategies utilized in the speculation of size areas, and, extra as a rule, to provide an effortless creation right into a huge number of geometrical issues concerning the proposal of distance, together with Riemannian and Carnot-Caratheodory metrics, the hyperbolic airplane, distance-volume inequalities, asymptotic geometry (large scale, coarse), Gromov hyperbolic areas, convergence of metric areas, and Alexandrov areas (non-positively and non-negatively curved spaces). The authors are inclined to paintings with "easy-to-touch" mathematical gadgets utilizing "easy-to-visualize" equipment. The authors set a tough aim of creating the middle components of the booklet obtainable to first-year graduate scholars. so much new innovations and techniques are brought and illustrated utilizing least difficult situations and heading off technicalities. The booklet includes many routines, which shape an integral part of exposition.
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Additional info for A Course in Metric Geometry (Graduate Studies in Mathematics, Volume 33)
In §4 of Chapter I we explained the relation of D to hyperbolic volumes. In particular, if M is any oriented compact hyperbolic 3-manifold (or complete hyperbolic 3-manifold with cusps), and if we triangulate M into oriented ideal hyperbolic tetrahedra ∆j , then the expression ξM = [zj ], where zj is the cross-ratio of the vertices of ∆j , lies in BC and the interpretation of D(zj ) as Vol(∆j ) implies that the imaginary part of Denh (ξM ) is the hyperbolic volume of M . The corresponding interpretation of the real part Denh (ξM ) ∈ R/π 2 Q is that it is equal (up to a normalizing factor) to the Chern-Simons invariant of M .
These series, about which there is a very extensive literature—with the letter “q” having been the traditional choice long before it was realized that there was any connection with the “q” of “quantum”—are functions of a formal (or small complex) variable q which are given by convergent inﬁnite series whose terms are rational functions of q with rational coeﬃcients. For instance, the q-hypergeometric functions, a very important subclass which includes some classical modular forms and related functions like Ramanujan’s “mock theta functions” (which have occurred in connection with quantum invariants of 3-manifolds ) are n given by series whose nth term has the form i=1 R(q, q i ) for some rational function R(x, y) of two variables.
63 The dilogarithm function, deﬁned in the ﬁrst sentence of Chapter I, is a function which has been known for more than 250 years, but which for a long time was familiar only to a few enthusiasts. In recent years it has become much better known, due to its appearance in hyperbolic geometry and in algebraic K-theory on the one hand and in mathematical physics (in particular, in conformal ﬁeld theory) on the other. I was therefore asked to give two lectures at the Les Houches meeting introducing this function and explaining some of its most important properties and applications, and to write up these lectures for the Proceedings.
A Course in Metric Geometry (Graduate Studies in Mathematics, Volume 33) by Dmitri Burago, Yuri Burago, Sergei Ivanov