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

By Jean Dieudonné

A vintage on hand back! This e-book strains the background of algebraic topology starting with its construction by means of Henry Poincaré in 1900, and describing intimately the real principles brought within the concept ahead of 1960. In its first thirty years the sphere appeared restricted to purposes in algebraic geometry, yet this replaced dramatically within the Nineteen Thirties with the construction of differential topology by way of Georges De Rham and Elie Cartan and of homotopy idea through Witold Hurewicz and Heinz Hopf. The impression of topology started to unfold to progressively more branches because it steadily took on a vital position in arithmetic. Written by way of a world-renowned mathematician, this e-book will make intriguing examining for someone operating with topology.

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Example text

Y = −(f + s)y + r, y(0) = 0. Consider any nontrivial solution x of the homogeneous equation x =−(1/2)(f +s)x. If u is the function satisfying u = 2r/x2 , u(0) = 0, then y = (1/2)ux2 . But u ≥ 0, so that y ≥ 0, and s ≤ f . This contradicts the fact that s is defined for all time, since f (t) → −∞ as t −1/s(0). The condition that the Jacobi fields in V span the normal space of c at every point is essential: Consider for example R2 with the standard metric, and a geodesic c in R2 . If E is a parallel field orthogonal to c, then t → J(t) := tE(t) defines a non-parallel Jacobi field, and represents an (n − 1)-dimensional space of Jacobi fields satisfying all the conditions of the theorem except for the one above.

Submersions, Foliations, and Metrics (vii) Denote by H : T M −→ H ⊂ T M the orthogonal projection onto the horizontal subbundle. It is interesting to note that the A and S tensors are just the covariant derivatives in horizontal and vertical directions of this projection restricted to H, cf. [80]. More precisely, for horizontal x and vertical u, ∇x H|H = Ax , Sx u = −(∇u H)x. In fact, for a horizontal field Y that evaluates to the vector y at the foot-point of x , ∇x (HY ) = (∇x H)Y − H∇x Y , so that (∇x H)Y = ∇x (HY ) − H∇x Y = Ax y, and a similar formula holds for ∇u H.

1, K(r) = 3 [X, Y ], T 4 2 = 3 . (1 + r2 )2 32 Chapter 1. 12) and summing over an orthonormal basis of the vertical space implies that for basic X, Y , 2 div AX Y = − tr d∇ SX,Y . 1, which asserts that the mean curvature κ of the foliation satisfies −dκ(X, Y ) = 2 div AX Y : since κ is just the trace of S, the above identity then follows from tr ◦ d∇ = d◦ tr, which itself reflects the fact that the trace operator is a parallel section of the bundle (End T M )∗ , see for example [136]. (vi) The reader may have noticed that some of the curvature formulae in this section seem to differ substantially from those in [22] or [97], even allowing for the difference in notation.

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