Is there a field of 'real analytic geometry'?

Question

I am wondering whether there is a field of 'real analytic geometry', and if not, why not? There are branches of geometry corresponding to increasingly large sets of functions: polynomial (algebraic geometry), analytic (complex geometry), differentiable (differential geometry), continuous (topology). 'Shapes' defined by analytic functions are studied in complex geometry, but as far as I can see only complex-analytic functions: is there nothing to study about figures defined by real-analytic functions?

Answer 1

Of course, there is an area called Real Analytic Geometry, dealing with real analytic spaces. It can be traced back to a seminal paper by H. Cartan (Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France 85, 1957, 77-99). One main difference in the real setting is the lack of coherence, a basic result in complex analytic spaces. A serious difficulty coming from this lack is that a set that is locally described by real analytic equations need not have global analytic equations. This lead to the introduction of global analytic or C-analytic sets by H. Whitney and F. Bruhat in Quelques propriétés fondamentales des ensembles analytiques réels (Comment. Math. Helv. 33,1959, 132-160), a fundamental paper. Another difficulty settled in this paper is the notion of irreducibility. In the complex setting this amounts to connectedness of the singular locus, while this topological condition is not enough for real analytic sets; also striking is that a real irreducible analytic set can consists of pieces of different dimensions. Thus we see from the very basis of the theory the peculiarities of the real category.

A somehow naive difference comes from the fact that reals have an order structure. Thus we can consider $\ge0$ and not only $=0$ to describe sets. This remark gives way to a whole new notion: semianalytic sets, introduced by S. Lojasiewicz (Ensembles semi-analytiques. I.H.E.S. Bures-sur-Yvette, 1964) in his studies of distributions. There is nothing like this in the complex realm! In the same vein, note that any system of real equations $f_1=\cdots=f_r=0$ can be replaced by the single equation $f_1^2+\dots+f_r^2=0$. Alas, is everything hypersurface? Not, the complex fact that one equation gives always a codimension 1 set fails over the reals: even the empty set can be described by a single equation ($x^2+1=0$). What is behind is the radical Nullstellensatz, again failing over the reals.

Also important, in the real setting there is no proper mapping theorem dealing with images of analytic sets. The failure of this important complex tool gives rise to a new class of sets, called subanalytic. They were introduced by H. Hironaka at the beginning of the 1970s; he studied them systematically using his desingularization theorems.

Third, it is worth remarking that in the real category everything is affine. Real projective spaces and real grassmannians can be analytically embedded in some $\mathbb{R}^n$, in fact, algebraically embedded. As a consequence, in the real category everything is Stein, that is, there are a plenty of analytic functions to do things. For instance, to represent objects from Algebraic Topology (homology, cohomology, homotopy classes) using analytic data.

One can consider also singularities of real analytic functions and maps as part of the field. Real Algebraic Geometry is an included area, but this is more formal than practical. In any case all these areas named REAL have a very strong connection with differential topology... as the Nash(-Tognoli) theorem quoted by @Matt E shows. And Nash functions are also a relevant subarea since M. Artin and B. Mazur called attention to them.

One point is that complex analytic structures are much more rigid than the corresponding real ones. On the other hand real analytic (or algebraic) structures include always the tool of complexification (as real numbers are the real part of complex numbers), through which one always analyses matters. Some people even say that real means just complex plus an involution (conjugation, so to say). In this sense all is a part of Complex Analytic Geometry, and any real expert will agree that seeing real objects as part of complex objects is always essential. All in all, there is a wealth of research literature in what we can call Real Analytic Geometry.

Answer 2

Yes, there is a theory of real analytic manifolds. However, it turns out that any smooth manifold can be promoted to a real analytic manifold in a unique way. See this MO question and answers for more details.

In fact, connected closed smooth manifolds can also be realized as Nash manifolds, which are between the algebraic and analytic worlds. See wikipedia for a brief discussion, and also p.91 of this paper of Artin and Mazur. (In case the link dies at some point, this is Artin and Mazur's paper On period points, in Annals of Math., vol. 81 (1965).)