Geometry and algebra can be hardly separated: “L'algèbre n'est qu'une géométrie écrite; la géométrie n'est qu'une algèbre figurée”. Geometric objects usually form algebras, such as Lie algebras of vector fields, associative algebras of differential operators, etc.

In this chapter we consider the associative algebra of differential operators on the projective line. Projective geometry allows us to describe this complicated and interesting object in terms of tensor densities. The group Diff+(S1) and its cohomology play a prominent role, unifying different aspects of our study.

The group Diff+(S1) of orientation-preserving diffeomorphisms of the circle is one of the most popular infinite-dimensional Lie groups connected to numerous topics in contemporary mathematics. The corresponding Lie algebra, Vect(S1), also became one of the main characters in various areas of mathematical physics and differential geometry. Part of the interest in the cohomology of Vect(S1) and Diff(S1) is due to the existence of their non-trivial central extensions, the Virasoro algebra and the Virasoro group.

We consider the first and the second cohomology spaces of Diff(S1) and Vect(S1) with some non-trivial coefficients and investigate their relations to projective differential geometry. Cohomology of Diff(S1) and Vect(S1) has been studied by many authors in many different settings; see [72] and [90] for comprehensive references. Why should we consider this cohomology here?

The group Diff(S1), the Lie algebra Vect(S1) and the Virasoro algebra appear consistently throughout this book.

On October 5, 2001, the authors of this book typed in the word “Schwarzian” in the MathSciNet database and the system returned 666 hits. Every working mathematician has encountered the Schwarzian derivative at some point of his education and, most likely, tried to forget this rather scary expression right away. One of the goals of this book is to convince the reader that the Schwarzian derivative is neither complicated nor exotic; in fact, it is a beautiful and natural geometrical object.

The Schwarzian derivative was discovered by Lagrange: “According to a communication for which I am indebted to Herr Schwarz, this expression occurs in Lagrange's researches on conformable representation ‘Sur la construction des cartes géographiques’” [117]; the Schwarzian also appeared in a paper by Kummer in 1836, and it was named after Schwarz by Cayley. The main two sources of current publications involving this notion are classical complex analysis and one-dimensional dynamics. In modern mathematical physics, the Schwarzian derivative is mostly associated with conformal field theory. It also remains a source of inspiration for geometers.

The Schwarzian derivative is the simplest projective differential invariant, namely, an invariant of a real projective line diffeomorphism under the natural SL(2,ℝ)-action on ℝℙ.

This chapter concerns projective geometry and projective topology of submanifolds of dimension greater than 1 in projective space. We start with a panorama of classical results concerning surfaces in ℝℙ3. This is a thoroughly studied subject, and we discuss only selected topics connected with the main themes of this book. Section 5.2 concerns relative, affine and projective differential geometry of non-degenerate hypersurfaces. In particular, we construct a projective differential invariant of such a hypersurface, a (1, 2)-tensor field on it. Section 5.3 is devoted to various geometrical and topological properties of a class of transverse fields of directions along non-degenerate hypersurfaces in affine and projective space, the exact transverse line fields. In Section 5.4 we use these results to give a new proof of a classical theorem: the complete integrability of the geodesic flow on the ellipsoid and of the billiard inside the ellipsoid. Section 5.5 concerns Hilbert's fourth problem: to describe Finsler metrics in a convex domain in projective space whose geodesics are straight lines. We describe integral-geometric and analytic solutions to this celebrated problem in dimension 2 and discuss the multi-dimensional case as well. The last section is devoted to Carathéodory's conjecture on two umbilic points on an ovaloid and recent conjectures of Arnold on global geometry and topology of non-degenerate closed hypersurfaces in projective space.

In this chapter we consider M, a smooth manifold of dimension n. How does one develop projective differential geometry on M? If M is a PGL(n + 1,ℝ)- homogeneous space, locally diffeomorphic to ℝℙn, then the situation is clear, but the supply of such manifolds is very limited. Informally speaking, a projective structure on M is a local identification of M with ℝℙn (without the requirement that the group PGL(n + 1, ℝ) acts on M). Projective structure is an example of the classic notion of a G-structure widely discussed in the literature. Our aim is to study specific properties of projective structures; see [121, 220] for a more general theory of G-structures.

There are many interesting examples of manifolds that carry projective structures; however, the general problem of existence and classification of projective structures on an n-dimensional manifold is wide open for n ≥ 3. There is a conjecture that every three-dimensional manifold can be equipped with a projective structure; see [196]. This is a very hard problem and its positive solution would imply, in particular, the Poincaré conjecture.

In this chapter we give a number of equivalent definitions of projective structures and discuss some of their main properties. We introduce two invariant differential operators acting on tensor densities on a manifold and give a description of projective structures in terms of these operators.

Lie algebras of vector fields on a smooth manifold M became popular in mathematics and physics after the discovery of the Virasoro algebra by Gelfand and Fuchs in 1967. Gelfand and Fuchs, Bott, Segal, Haefliger and many others studied cohomology of Lie algebras of vector fields and diffeomorphism groups with coefficients in spaces of tensor fields. This theory attracted much attention in the last three decades, many important problems were solved and many beautiful applications, such as characteristic classes of foliations, were found.

In this chapter we consider cohomology of Lie algebras of vector fields and of diffeomorphism groups with coefficients in various spaces of differential operators; this is a generalization of Gelfand–Fuchs cohomology. Only a few results are available so far, mostly for the first cohomology spaces. The main motivation is to study the space of differential operators Dλ,μ(M), viewed as a module over the group of diffeomorphisms.

This cohomology is closely related to projective differential geometry and, in particular, to the Schwarzian derivative. The classic Schwarzian derivative is a 1-cocycle on the group Diff(S1), related to the module of Sturm–Liouville operators. Multi-dimensional analogs of the Schwarzian derivative are defined as projectively invariant 1-cocycles on diffeomorphism groups with values in spaces of differential operators.

By now, even a skeptical reader should be thoroughly sold on the utility of embeddings into cubes. But the sales pitch is far from over! We next pursue the consequences of a result stated earlier: If X is completely regular, then Cb(X) completely determines the topology on X. In brief, to know Cb(X) is to know X. Just how far can this idea be pushed? If Cb(X) and Cb(Y) are isomorphic (as Banach spaces, as lattices, or as rings), must X and Y be homeomorphic? Which topological properties of X can be attributed to structural properties of Cb(X) (and conversely)?

These questions were the starting place for Marshall Stone's 1937 landmark paper, “Applications of the Theory of Boolean Rings to General Topology” [140]. It's in this paper that Stone gave his account of the Stone– Weierstrass theorem, the Banach–Stone theorem, and the Stone– Čech compactification. (These few are actually tough to find among the dozens of results in this mammoth 106-page work.) A signal passage from his introduction may be paraphrased as follows: “We obtain a reasonably complete algebraic insight into the structure of Cb(X) and its correlation with the structure of the underlying topological space.” Stone's work proved to be a gold mine – the digging continued for years! – and its influence on algebra, analysis, and topology alike can be seen in virtually every modern textbook.

Independently, but later that same year (1937), Eduard Čech [24] gave another proof of the existence of the compactification but, strangely, credits a 1929 paper of Tychonoff for the result (see Shields [136] for more on this story).

As pointed out earlier, the spaces Lp and lp exhaust the “isomorphic types” of Lp(μ) spaces. Thus, to better understand the isomorphic structure of Lp(μ) spaces, we might ask, as Banach did:

For p ≠ r, can lr or Lr be isomorphically embedded into Lp?

We knowquite a bit about this problem.We knowthat the answer is always yes for r = 2, and, in case 2 < p < ∞, the Kadec–Pelczyński theorem (Corollary 9.7) tells us that r = 2 is the only possibility. In this chapter we'll prove the following statement:

If p and r live on opposite sides of 2, there can be no isomorphism from Lr or lr into Lp.

This leaves open only the cases 1 = r < p < 2 and 1 ≤ p < r < 2. The first case can also be eliminated, as we'll see, but not the second.

Unconditional Convergence

We next introduce the notion of unconditional convergence of series. What follows are several plausible definitions.

We say that a series Σn xn in a Banach space X is

1. (a) unordered convergent if Σn xπ(n) converges for every permutation (one–to–one, onto map) π : ℕ → ℕ;

2. (b) subseries convergent if Σk xnk converges for every subsequence (xnk) of (xn);

3. (c) “random signs” convergent if Σn εn xn converges for any choice of signs εn = ±1;

4. (d) bounded multiplier convergent if Σn an xn converges whenever |an| ≤ 1.