Download A Course in Metric Geometry (Graduate Studies in by Dmitri Burago, Yuri Burago, Sergei Ivanov PDF

By Dmitri Burago, Yuri Burago, Sergei Ivanov

"Metric geometry" is an method of geometry according to the proposal of size on a topological house. This strategy skilled a truly quickly improvement within the previous few many years and penetrated into many different mathematical disciplines, akin to staff thought, dynamical platforms, and partial differential equations. the target of this graduate textbook is twofold: to offer an in depth exposition of simple notions and strategies utilized in the speculation of size areas, and, extra ordinarily, to provide an simple advent right into a vast number of geometrical subject matters on the topic of the concept of distance, together with Riemannian and Carnot-Caratheodory metrics, the hyperbolic aircraft, 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 likely to paintings with "easy-to-touch" mathematical items utilizing "easy-to-visualize" equipment. The authors set a tough target of constructing the middle components of the e-book available to first-year graduate scholars. such a lot new thoughts and strategies are brought and illustrated utilizing easiest circumstances and warding off technicalities. The ebook includes many routines, which shape an integral part of exposition.

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Additional resources for A Course in Metric Geometry (Graduate Studies in Mathematics, Volume 33)

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Aw~qAw 76 q+IA'-'Aw in . 8) we have (-i) q Hq(V) do = ~ - ~ - ~J........ '''qp ! '"(qp-l)!! I p(p+2) "'" (p+q-2) ' if all of q~are even, [ O , otherwise. '''~P! +~p=~ ~(2DI'''''2DP) = K ~ d o ~i ! NN! 5) holds. 11). 4) we have m-i r=O (-l)r (m-l" [fc (6) ]m-l-r [gc (6) ]rMc (~:) r ) = .

The r-th elementary symmetric function of the principal curvatures of Ng divided by its term number (m-l. r )" We assume that HO=I. Hq(V)doAd0p_ 1 , where H (v) is the q-th mean curvature of N with respect to the normal vector v. Weyl ([6],[7]), but here we give a brief proof different from theirs. Lemma 4 At each point x ~ N, the average of H (v) when v passes over the whole q unit sphere Sp-I in the normal subspace Vx(N) is Kc 2D i I ~ J H (v)dO Op_ 1 s P _ l q = , if q=2p, ( 2H) n p(p+2) •..

O2p "''Kq %li2JlJ2 - . 6) Kc ijkl = Rijkl c(6ik6jl-6il6jk )" Rijkl is the curvature tensor of N. 5) should be understood to be I. -. - A (W~+~x%~) q! q! -. qp! I.. (%m~Pf~(q 1 •. 8) 50 ~(ql'" "'qp) = iq 61 .... n " $I .... /%0~q-qp+iA. Aw~qAw 76 q+IA'-'Aw in . 8) we have (-i) q Hq(V) do = ~ - ~ - ~J........ '''qp ! '"(qp-l)!! I p(p+2) "'" (p+q-2) ' if all of q~are even, [ O , otherwise. '''~P! +~p=~ ~(2DI'''''2DP) = K ~ d o ~i ! NN! 5) holds. 11). 4) we have m-i r=O (-l)r (m-l" [fc (6) ]m-l-r [gc (6) ]rMc (~:) r ) = .

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