# Some Notions of Local Differential Topology

## Abstract

Exaggerating only slightly, one can say that local differential (or local analytic) topology is nothing but the art of making clever use of the implicit function theorem,† or equivalently, its simpler version called the inverse mapping theorem, which we shall recall to start with. But first let us agree on the following conventions, meant to simplify the exposition: We shall deal with mappings of a Euclidean space (called the source) into another Euclidean space (called the goal); it will not be necessary for these mappings to be defined in the whole of the source but only in some domain—which we shall not specify explicitly. The coordinates of a point in the source will be denoted by x i, those of a point in the goal by y j. A mapping f: ℝm → ℝn is therefore defined by
$$y_{1} = f_{1}\left (x_{1},\ x_{2},\ ...,\ x_{m}\right);\ y_{2} = f_{2}\left (x_{1},\ x_{2},\ ...,\ x_{m};\right);\ ...\ y_{n} = f_{n}\left (x_{1},\ x_{2},\ ...,\ x_{m}\right)$$
where f 1, f 2, …, f n are n real, numerical functions defined in some common domain of ℝm. If these functions are differentiable (respectively, analytic) in that domain, we shall say that the mapping is differentiable (respectively, analytic). It is understood that “differentiable” means r times continuously differentiable, where r is some integer (1 ≤ r ≤ ∞) chosen once for all.

## Keywords

Euclidean Space Jacobian Matrix Canonical Projection Transversality Condition Tangent Mapping

## Bibliographic Annotation

1. An exposition of the basic concepts of differential topology can be found in the beginning of the book of S. Sternberg, “Lectures on Differential Geometry,” Prentice-Hall Mathematics Series, Prentice Hall, (1964).Google Scholar
2. Concerning the nondenumerable infinity of topological types of polynomial mappings, see R. Thorn, “L’Enseignement mathématique,” VII: 24–33 (1962).Google Scholar
3. For the notion of “generic type,” and the definition of S k types, S k(S k) types, etc., see R. Thorn, Ann. Inst. Fourier, 6: 43–87 (1956).
4. The special case when the goal is one-dimensional constitutes “Morse Theory” on which a very illuminating book is available: J. Milnor, “Morse Theory,” (Ann. of Math. Studies No. 51) Princeton University Press, (1963).Google Scholar
5. The mappings of the plane into the plane are investigated in detail by H. Whitney, Ann. of Math. 62 (3): 374–410 (1965).

## Authors and Affiliations

• F. Pham
• 1
• 2
1. 1.C.E.N.SaclayFrance
2. 2.C.E.R.N.GenevaSwitzerland