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

By Dmitri Burago, Yuri Burago, Sergei Ivanov

"Metric geometry" is an method of geometry in line with the thought of size on a topological house. This procedure skilled a really speedy improvement within the previous couple of many years and penetrated into many different mathematical disciplines, similar to staff idea, dynamical structures, and partial differential equations. the target of this graduate textbook is twofold: to offer a close exposition of simple notions and methods utilized in the idea of size areas, and, extra in most cases, to supply an trouble-free creation right into a wide number of geometrical issues concerning the suggestion 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 likely to paintings with "easy-to-touch" mathematical gadgets utilizing "easy-to-visualize" tools. The authors set a not easy target 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 easiest circumstances and warding off technicalities. The ebook includes many routines, which shape an essential component of exposition.

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And it is called mixed totally geodesic if its second fundamental form σ satisfies σ(X, Z) = 0 for any X ∈ D and Z ∈ D⊥ . A mixed totally geodesic CR-submanifold is called mixed foliate if its complex distribution D is also integrable. Obviously, real hypersurfaces of a Kähler manifold are anti-holomorphic submanifolds with p = rank D⊥ = 1. ˜ m (4c) with c = 0 admits no mixed foliate proper Lemma 4 A complex space form M CR-submanifolds. -Y. Chen Lemma 4 is due to [4] for c > 0 and due to [29] for c < 0.

2h + 1. 51) j=1 for i = 1, . . , 2h. 18) and the equation of Gauss that 2h 2h σi,i σ2h+1,2h+1 − δ(D) = i=1 (σi,2h+1 )2 + 2hc. 52) i=1 On the other hand, we have 2h σi,i σ2h+1,2h+1 = i=1 (2h + 1)2 2 1 − → H − (σ2h+1,2h+1 )2 − 2h2 | H D |2 . 53) we obtain δ(D) = (2h + 1)2 2 1 − → H + 2hc − 2h2 | H D |2 − (σ2h+1,2h+1 )2 2 2 2h − (σi,2h+1 )2 i=1 (2h + 1)2 2 H + 2hc. -Y. 48) holds identically if and only if the following two statements hold: (i) N is a special Hopf hypersurface and ˜ h+1 (4c). (ii) N is D-minimal in M Obviously, conditions (i) and (ii) imply that N is a minimal real hypersurface of ˜ h+1 (4c).

30). 46) identically. 4. By condition (a), N is D-minimal. 47) for any unit vector e1 ∈ D. 47) and polarization that the second fundamental form satisfies the following condition: σ(X, JY ) = σ(JX, Y ), ∀X, Y ∈ D. Therefore, according to Lemma 2(1), we may conclude that D is integrable. 5. 8 An Optimal Inequality for Real Hypersurfaces Obviously, anti-holomorphic submanifolds with rank D⊥ = 1 are nothing but real ˜ is called a Hopf hyperhypersurfaces. , an eigenvector of the shape operator Aξ , where ξ is a unit normal vector of N.

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