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.

**Read Online or Download A History Of Algebraic And Differential Topology, 1900-1960 PDF**

**Similar differential geometry books**

Tight and taut manifolds shape a tremendous and distinctive classification of surfaces inside differential geometry. This booklet includes in-depth articles by means of specialists within the box in addition to an in depth and accomplished bibliography. This survey will open new avenues for additional examine and should be a tremendous addition to any geometer's library.

**The geometry of Kerr black holes**

This specific monograph by way of a famous UCLA professor examines intimately the maths of Kerr black holes, which own the houses of mass and angular momentum yet hold no electric cost. appropriate for complex undergraduates and graduate scholars of arithmetic, physics, and astronomy in addition to specialist physicists, the self-contained remedy constitutes an advent to trendy innovations in differential geometry.

Supplying an updated assessment of the geometry of manifolds with non-negative sectional curvature, this quantity provides a close account of the latest examine within the quarter. The lectures conceal quite a lot of issues resembling basic isometric workforce activities, circle activities on absolutely curved 4 manifolds, cohomogeneity one activities on Alexandrov areas, isometric torus activities on Riemannian manifolds of maximal symmetry rank, n-Sasakian manifolds, isoparametric hypersurfaces in spheres, touch CR and CR submanifolds, Riemannian submersions and the Hopf conjecture with symmetry.

**The Principle of Least Action in Geometry and Dynamics**

New variational tools via Aubry, Mather, and Mane, chanced on within the final 20 years, gave deep perception into the dynamics of convex Lagrangian platforms. This e-book exhibits how this precept of Least motion seems to be in various settings (billiards, size spectrum, Hofer geometry, glossy symplectic geometry).

- Manifolds and Geometry
- Differential Geometry
- Gnomon: from pharaohs to fractals
- Collected papers on Ricci flow
- Dynamical Systems (Colloquium Publications)

**Additional info for A History Of Algebraic And Differential Topology, 1900-1960**

**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.