# Download A History Of Algebraic And Differential Topology, 1900-1960 by Jean Dieudonné PDF

By Jean Dieudonné

A vintage on hand back! This booklet lines the historical past of algebraic topology starting with its construction by means of Henry Poincaré in 1900, and describing intimately the $64000 principles brought within the concept ahead of 1960. In its first thirty years the sector appeared constrained to functions in algebraic geometry, yet this replaced dramatically within the Nineteen Thirties with the production of differential topology through Georges De Rham and Elie Cartan and of homotopy concept via Witold Hurewicz and Heinz Hopf. The effect of topology started to unfold to increasingly more branches because it progressively took on a vital position in arithmetic. Written by means of a world-renowned mathematician, this e-book will make interesting interpreting for somebody operating with topology.

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Then the sectional curvatures of the coordinates horosphere t = const in the corresponding point and direction is equal to γ2 e2k2 t . From Gauss’ formula we get that the sectional curvature of the ambient space M n+1 at the same direction is equal to γ2 e2k2 t − k22 , −∞ ≤ t < +∞. As the sectional curvature M n+1 satisﬁes the inequality −k12 ≥ K ≥ −k22 , it follows that γ2 = 0 and the manifold M n+1 is a space of constant curvature −k22 . Proof of Theorem 2. From the part I) of theorem 1 it follows that F n is a compact convex hypersurface diffeomorphic to S n .

Let η be the 1-form on T1 M dual to ξ with respect to gS , and ϕ the (1,1) tensor given by ϕ X = J X − η (X)N . Then, (ξ, η, ϕ, g) ¯ = 1 1 η , 2ξ , ϕ , gS 2 4 is the standard contact metric structure on T1 M. Note that the contact metric of T1 M ¯ and (T1 M, gS ) share is given by g¯ = 41 gS . So, since g¯ is homothetic to gS , (T1 M, g) many geometric properties. For example, the former is reducible if and only if the latter is, and (T1 M, g) ¯ is locally symmetric, respectively semi-symmetric, if and only if (T1 M, gS ) has the same property.

From (1), we have hϕe = −λϕe. We can refer to [CPV] for more details. The components of the Ricci operator Q, with respect to {ξ, e, ϕe}, are given by (see [P3] or [CPV]) ⎧ 2 ⎪ ⎨ Qξ = 2(1 − λ )ξ + Ae + Bϕe, (13) Qe = Aξ + 2r − 1 + λ2 + 2aλ e + ξ(λ)ϕe, ⎪ ⎩ r 2 Qϕe = Bξ + ξ(λ)e + 2 − 1 + λ − 2aλ ϕe. Starting from (12) and using (13), one can characterize three-dimensional semisymmetric contact metric manifolds by a list of algebraic formulas involving λ, a, A, B: Lemma 1. [CP2] Let (M, η, g) be a three-dimensional non-Sasakian contact metric manifold.