By Dmitri Burago, Yuri Burago, Sergei Ivanov

"Metric geometry" is an method of geometry in accordance with the proposal of size on a topological house. This method skilled a truly quick improvement within the previous few many years and penetrated into many different mathematical disciplines, akin to crew conception, dynamical structures, and partial differential equations. the target of this graduate textbook is twofold: to offer a close exposition of easy notions and strategies utilized in the idea of size areas, and, extra normally, to supply an ordinary creation right into a vast number of geometrical subject matters concerning 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 hard aim of creating the middle elements of the ebook available to first-year graduate scholars. such a lot new ideas and strategies are brought and illustrated utilizing easiest circumstances and heading off technicalities. The booklet comprises many routines, which shape an essential component of exposition.

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

**Example text**

X ne ) : U → R is any smooth extension of X on U, we have div X = δh X he on ∂ E. 29. 9) it follows that the canonical extension of the unit normal vector ﬁeld to ∂ E pointing toward the interior of Rn \ E is given by ∇d(z), z ∈ U. Hence the deﬁnition of tangential divergence implies that d = div ∇d on ∂ E. 45) Therefore the mean curvature vector of ∂ E can be written as div ∇d ∇d. 30 (Tangential Laplacian). Let u ∈ C ∞ (∂ E). The tangenu of u on ∂ E is deﬁned as tial Laplacian u := div ∇ u on ∂ E.

10 (Normal velocity using the level sets). The normal velocity vector can also be expressed as follows. Let u : [a, b] × Rn → R be a continuous function of class C ∞ in [a, b] × A, where A ⊂ Rn is an {u(t, ·) = 0}, and such that open set containing t∈[a,b] u 2 + |∇u|2 > 0 in [a, b] × A. 12) Then the map f : [a, b] → P(Rn ) deﬁned as f (t) := {z ∈ Rn : u(t, z) ≤ 0}, t ∈ [a, b], is a smooth ﬂow, and ∂ f (t) = {z ∈ Rn : u(t, z) = 0}, (8) See [15, Theorem 8]. t ∈ [a, b]. 13) 42 Giovanni Bellettini The normal velocity vector equals − ∂u ∂t ∇u |∇u| |∇u| on ∂ f (t).

26 (Tangential gradient). Let u ∈ C ∞ (∂ E). The vector ﬁeld δu restricted to ∂ E is called the tangential gradient of u on ∂ E, and is denoted by ∇ u. It is not difﬁcult to show that the tangential gradient of u coincides on ∂ E with the δ operator applied to any smooth extension of u in U(11) : indeed, if u1 and u2 are two smooth extensions of u in U, then ∂ E ⊆ {u1 − u2 = 0}, so that ∇(u1 − u2 ) = ∇(u1 − u2 ), ∇d ∇d on ∂ E. Hence ∇ (u1 − u2 ) = 0 on ∂ E. 44) so that ∇u = δu in U. (10) For clarity of exposition, in this chapter we have tried to distinguish a function deﬁned on ∂ E from a function deﬁned in a neighbourhood U of ∂ E, using two different symbols (u and u respectively).