By Dmitri Burago, Yuri Burago, Sergei Ivanov

ISBN-10: 0821821296

ISBN-13: 9780821821299

"Metric geometry" is an method of geometry in keeping with the idea of size on a topological area. This procedure skilled a really speedy improvement within the previous few a long time and penetrated into many different mathematical disciplines, corresponding to workforce concept, dynamical platforms, and partial differential equations. the target of this graduate textbook is twofold: to provide a close exposition of easy notions and strategies utilized in the speculation of size areas, and, extra quite often, to supply an undemanding advent right into a large number of geometrical themes relating to the proposal 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 inclined to paintings with "easy-to-touch" mathematical gadgets utilizing "easy-to-visualize" tools. The authors set a demanding aim of creating the middle components of the e-book available to first-year graduate scholars. so much new innovations and techniques are brought and illustrated utilizing least difficult circumstances and heading off technicalities. The e-book includes many workouts, which shape an integral part of exposition.

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

Example text

1 1 1 ........................................... .............. 20) ... But by construction these groups R ExtiR (N,M ) are also functorial in N if we fix M , the functors N −→ R Exti (N,M ) are contravariant. Analogously we choose a projective resolution P• −→ N −→ 0 and define L Ext•R (N,M ) = H• (HomR (P• ,M )).

M ........................................... I 0 ........................................... M ............................. M I 0⊕I ........................................... 1 and W → I 1 0 I 0 We have U → I 0 ............................ 0 V .......................................... 0 ............................................ 0 ... ... ... .. .. W ... ... ... .. .. ... ... ... .......... ... ... ... .. .. U ... ... ...

3. An acyclic resolution of M ∈ Ob(ModΓ ) by objects in C is an exact sequence of Γ-modules 0 −→ M −→ X 0 −→ X 1 −→ X 2 −→ · · · where the X ν ∈ C. 4. If C is a class of acyclic objects for the derived functor {M −→ H i (Γ,M )}, and if 0 −→ M −→ X 0 −→ X 1 −→ · · · is an acyclic resolution of M by objects in C, then we have an isomorphism H i ((X • )Γ ). H i (Γ,M ) Proof: By induction on i. For i = 0 we get the exact sequence 0 −→ M Γ −→ (X 0 )Γ −→ (X 1 )Γ −→ . . and M Γ −→ ker((X 0 )Γ −→ (X 1 )Γ ) = H 0 ((X • )Γ ).

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A Course in Metric Geometry (Graduate Studies in Mathematics, Volume 33) by Dmitri Burago, Yuri Burago, Sergei Ivanov


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