http://euro-math-soc.eu/review/introduction-geometry-and-topology

This book is an excellent companion to everyone involved in a somehow advanced course on Differential Topology and/or Differential Geometry. Either students or professors will find it profitable, although students should have some previous acquaintance with the topics covered. In general, the author gets quite efficiently without diversions to several deep results, with a most friendly style. In some cases he does not try the impossible, and stops where mathematics get too tough, and there he proposes suitable references. Also, we find a selection of problems enough to make the book a valuable reference. The first chapter on Topology is a kind of (good) speedy introduction to the matter, that one should better classify as a recall. But then there is a beautiful presentation of the Jordan Curve theorem. This is worth by itself, and furthermore provides motivation for the posterior proof of the higher dimension version using de Rham cohomology. The second chapter is a terse presentation of the notions of differentiable manifolds. However terse, it yet succeeds in doing everything not embedded and very readable. Even vector bundles are defined in general form. True that this chapter does not get much beyond definitions (this reviewer for instance misses some words on flows), but anyway it offers a good introductory view of many things. Lie groups for instance. The third chapter would alone justify any previous omisions. This is a neat, motivating and accomplished little first course on de Rham Cohomology. A beauty. A lot is motivated, a lot is explained and a lot is distilled: Poincaré Lemma, Mayer-Vietoris, cohomology of spheres, Brouwer fixed point theorem, Jordan-Brouwer separation theorem… The last chapter on Differential Geometry is truly nice. It gives an account on curves form a higher viewpoint that is a bonus in form and content even for those who already know the matter. Then the author turns to curves in immersed manifolds: first fundamental form and internal geometry. Next, exterior geometry and the second fundamental form. Here the subject are hypersurfaces, enlarging the usual restriction to surfaces in 3-space. In the end, we get to Gauss Egregium theorem, nothing better in closing. It is clear that the book comes from the experience in the class room, and a good one it seems. We must thank the author for sharing it with us.