# Guide Differential Topology

This is a text designed for a real course on Differential Topology for postgraduate students.

## Elements of Differential Topology

The topics are as expected: smooth manifolds, tangent spaces and derivatives, regular values and transversality, vector bundles, tangent fields and flows and some global basic facts. There are many examples carefully chosen and described for motivation, as well as many exercises with hints to help the student. A very good book on the matter. Skip to main content. Login Register. I have just finished my undergraduate studies. During last two semesters I've taken two subjects dealing with manifolds:.

## Differential Topology 19

Analysis on manifolds , containing: definition of manifold, tangent space as derivations and classes of curves , vector fields, vector bundles, flows, Lie derivatives, integration on manifolds Stokes theorem , forms, Hodge decomposition theorem. Introduction to differential geometry for me it should be called introduction to Riemannian manifolds containing: tensor calculus introduction, Riemannian manifold definition, connections, curvatures, geodesic, normal coordinates, geodesic completness theorem, classification throught curvature, jacobi fields, harmonic maps.

On the other hand I am preparing for taking part in local conference called: "algebraic and differential topology". I have read: I- books references , II- books references , III- wikipedia , yet I still don't really know if differential topology is subtheory of differential geometry or is it seperate theory and how is it located between differential geometry and algebraic topology.

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For example I expect that studies on Riemmanian manifolds are part of differential geometry but would problem of classification manifolds up to diffeomorphism be a part of differential topology or geometry? Request : I would be grateful for your characterisation of differential topology and differential geometry possibly with examples of problems, theorems present at them.

First of all, the concept of a "manifold" is certainly not exclusive to differential geometry. Manifolds are one of the basic objects of study in topology, and are also used extensively in complex analysis in the form of Riemann surfaces and complex manifolds and algebraic geometry in the form of varieties. Within topology, manifolds can be studied purely as topological spaces, but it is also common to consider manifolds with either a piecewise-linear or differentiable structure.

The topological study of piecewise-linear manifolds is sometimes called piecewise-linear topology , and the topological study of differentiable manifolds is sometimes called differential topology. I'm not sure I would necessarily describe these as distinct subfields of topology -- they are more like points of view towards geometric topology, and for the most part one can study the same geometric questions from each of the three main points of view.

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However, there are questions that only make sense from one of these points of view, e. Differential topology can be found in position 57Rxx on the Math Subject Classification. Differential geometry, on the other hand, is a major field of mathematics with many subfields. It is concerned primarily with additional structures that one can put on a smooth manifold, and the properties of such structures, as well as notions such as curvature, metric properties, and differential equations on manifolds.

Classical differential geometry, i.

Smale , On gradient dynamic systems , Annals of Math. MR 24 A Zbl Stallings , The topology of high-dimensional piece-wise linear manifolds , to appear. Whitehead , Simplicial spaces , nuclei and m-groups, Proc.

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London Math. JFM Whitehead , Simple homotopy types , Amer.

MR 11,c Zbl Whitehead , C1-complexes , Annals of Math. Whitney , Differentiable manifolds , Amer.