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Projective differential geometry old and new. From the Schwarzian derivative to the cohomology of diffeomorphism groups. (English) Zbl 1073.53001

Cambridge Tracts in Mathematics 165. Cambridge: Cambridge University Press (ISBN 0-521-83186-5). xi, 249 p. (2005).
The guiding theme of this book, the Schwarzian derivative \({\mathbf S}(f)=f'''/f'-{3 \over 2}(f''/f')^2\), is the oldest differential invariant of projective geometry, dating back to times of Lagrange, Kummer and Cayley. It tells whether two functions \(f_1,f_2\) of one variable are related by a fractional linear transformation \(f_2=(a\,f_1+b)/ (c\,f_1+d)\). By this, it classifies all projective structures on the projective line \({\mathbb R}{\mathbb P}^1\), considered as diffeomorphisms of \({\mathbb R}{\mathbb P}^1\) or as curves in \({\mathbb R}{\mathbb P}^1\). This is one of a number of issues of S that indicate possible generalizations to higher dimensions.
A second such issue is the following identification of the space of projective structures with the space of Sturm-Liouville operators: Consider a local identification of such diffeomorphisms \(\gamma(t)\) with functions \(f(t)\) obtained by choosing a curve \(\Gamma(t)\) in \({\mathbb R}^2\) representing \(\gamma(t)\) in an affine chart. Imposing the condition, that the area \(A_\Gamma\) spanned by the vectors \(\Gamma(t)\) and \(\Gamma'(t)\) is 1, \(\Gamma(t)\) is almost unique by this demand and of the form \(\Gamma(t)= (f'(t)^{-1/2},f(t)\,f'(t)^{-1/2})\). From \(A_\Gamma=1\) one infers that \(\Gamma(t)\) and \(\Gamma''(t)\) are proportional, i.e., \(\Gamma''(t)+u(t) \Gamma(t)=0\) and a computation shows that \(u={\mathbf S}(f)\). In a similar way, curves \(\gamma(t)\) in \({\mathbb R}{\mathbb P}^n\) are identified with linear differential operators \(A\) on \(S^1\) of order \(n+1\).
As a third issue a link to group cohomology is at hand: The chain rule of S reads \({\mathbf S}(f\circ g)={\mathbf S}(f)\circ g+{\mathbf S}(g)\). This reveals the Schwarzian derivative as a cocycle of the diffeomorphism group \(\text{Diff}(S^1)\) of the 1-sphere with coefficients in the module \({\mathcal F}_2({\mathbb R}{\mathbb P}^1)\) of quadratic differentials.
In section 1, the Schwarzian derivative \({\mathbf S}(f)\) is introduced as a measure, how a diffeomorphism \(f:{\mathbb R}{\mathbb P}^1\to{\mathbb R}{\mathbb P}^1\) changes the cross ratio. It is shown that it is a unique \(\text{PGL}(n,{\mathbb R})\)-relative 1-cocycle of \(\text{Diff} ({\mathbb R}{\mathbb P}^1)\) with values in \({\mathcal F}_2\). Considering the Virasoro algebra \(\text{Vir}={\text{Vect}}(S^1)\oplus{\mathbb R}\) and an explicit formula for its coadjoint action on its regular dual space \({\text{Vir}}_{\text{reg}}^\ast=C^\infty(S^1)\oplus{\mathbb R}\) one obtains a remarkable identification of \({\text{Vir}}_{\text{reg}}\) with the space of Sturm-Liouville operators, therefore with projective structures.
Sections 2 and 3, entitled ‘Geometry of the projective line’ and ‘Algebra of the projective line and cohomology of \(\text{Diff} ({\mathbb R}{\mathbb P}^1)\)’, respectively, are concerned with notions and results bringing together notions of projective differential geometry and several algebraic constructions. The simplest invariants of curves in \({\mathbb R}{\mathbb P}^n\) are the projective curvature and the cubic form. Homotopy of non-degenerate curves is discussed and relations between projectively invariant differential operators and differential invariants of curves in \({\mathbb R}{\mathbb P}^n\) are established. The inverse of the symbol map is a \(\text{PGL}(n,{\mathbb R})\)-equivariant isomorphism from the filtered algebra \({\mathcal D}_{\lambda,\mu}(S^1)\) of linear differential operators between tensor densities \({\mathcal F}_\lambda, {\mathcal F}_\mu\) of weights \(\lambda\) and \(\mu\), respectively, to tensor densities on \(S^1\). It gives a projectively equivariant quantization map. Special interest is devoted to transvectants, i.e., certain bilinear differential operators \(J^{\lambda,\mu}_m\) from \({\mathcal F}_\lambda\otimes{\mathcal F}_\mu\) to \({\mathcal F}_{\lambda+\mu+m}\).
Next, global results, generalizing the classical 4-vertex theorem for curves in the Euclidean plane, are treated. Section 4 (‘Vertices of projective curves’), is concerned with an important global result of projective differential geometry of curves, namely Barner’s theorem stating that each closed, strictly convex curve in \({\mathbb R}{\mathbb P}^n\) has at least \(n+1\) inflection points [M. Barner, Über die Mindestanzahl stationärer Schmiegebenen bei geschlossenen strengkonvexen Raumkurven. Abh. Math. Semin. Univ. Hamb. 20, 196–215 (1956; Zbl 0071.15601)]. A 3-dimensional version of this is the 6-vertex theorem: Any closed convex curve in \({\mathbb R}{\mathbb P}^2\) has at least 6 sextactic points, i.e., points, where it has 6th-order contact with its osculating conic. These results are in close relation to a theorem of E. Ghys: The Schwarzian derivative \({\mathbf S}(f)\) of a diffeomorphism \(f\) of \({\mathbb R}{\mathbb P}^1\) vanishes in at least 4 points. Discrete versions of these notions and results are concerned with polygons \(X\) in \({\mathbb R}{\mathbb P}^n\). Much of the material concerned with discretization refers to work of the authors [see for instance, V. Ovsienko, S. Tabachnikov, Enseign. Math., II. Sér. 47, No.1–2, 3–19 (2001; Zbl 1060.53010)].
In section 5, one is ascending from curves towards submanifolds of \({\mathbb R}{\mathbb P}^n\) of arbitrary dimension \(k\). In the case \(k=2, n=3\), following early work E. F. Wilczynski [Projective differential geometry of curved surfaces (memoirs 1–5), Am. M. S. Trans. (8), 233–260 (1907; JFM 38.0633.03); (9), 79–120 (1908; JFM 39.0671.02); (9), 293–315 (1908; JFM 39.0671.03); (10), 176–200 (1909; JFM 40.0663.03); (10), 279–296 (1909; JFM 40.0663.04)], certain natural asymptotic coordinates \(u, v\) on a surface \(M\subset{\mathbb R}{\mathbb P}^3\), are considered. This reduces the group of symmetries of the tangent plane from \(\text{PGL}(2,{\mathbb R})\) to \(\text{PGL}(1,{\mathbb R})\times\text{PGL}(1,{\mathbb R})\). One obtains invariants \(\alpha, \beta\) arising as polarized tensor densities of degrees \((\mu,\nu)=(-1,2)\) and \((\mu,\nu)=(2,-1)\) respectively, i.e., as sections of line bundles \({\mathcal F}_{\lambda,\mu}(M)= (L_u^\ast)^{\otimes\lambda}\otimes (L_v^\ast)^{\otimes\mu}\), where \(L_u^\ast, L_v^\ast\) are dual line bundles defined by the coordinates \(u,v\). Similar to the case of curves, these invariants are in relation to Sturm-Liouville operators, depending on \(u\) and \(v\) respectively. In higher dimension, relative, affine, and projective differential geometry of hypersurfaces is discussed. Global topics treated in this chapter are complete integrability of the geodesic flow (special cases of ellipsoids and the billiard map inside the ellipsoid) bisymplectic maps, projective billiards, Hilbert’s fourth problem, Carathéodory’s conjecture on umbilic points.
Section 6 deals with global questions about flat projective structures on smooth manifolds \(M^n\). Following ideas of J. M. C. Whitehead and Ch. Ehresmann these are defined by maps from the universal covering of \(M^n\) to \({\mathbb R}{\mathbb P}^n\), equivariant with respect to a monodromy homomorphism \(T\) from the fundamental group \(\pi_1(M)\) to \(\text{PGL}(n+1,{\mathbb R})\), so-called developing maps. Projective structures induce a rich geometry on \(M\), for instance, geodesics and geodesic submanifolds can be defined as pre-images of lines and projective subspaces of \({\mathbb R}{\mathbb P}^n\) under the developing map. A multi-dimensional analogon of Sturm’s theorem on zeroes is indicated, the case \(n=2\) of which asserts that for three connected geodesics \(\gamma_1, \gamma_2, \gamma_3\) on a simply connected surface \(M^2\) with a projective structure the intersection points of \(\gamma_1\) with \(\gamma_2\) and \(\gamma_3\) alternate. Two invariant differential operators on tensor densities and a description of projective structures in terms of these operators are given. Relations between projective and contact structures are discussed. In dimension \(n=2\), projective structures are classified.
Finally, section 7 introduces \(n\)-dimensional versions of the Schwarzian derivative as projectively invariant cocycles on diffeomorphism groups of \(n\)-dimensional manifolds \(M\) with values in spaces of differential operators. One of these invariants \({\mathcal L}\) is defined by assigning to a projective connection on \(M\) a map \(L:{\text{ Diff}}^\infty(M)\to C^\infty({\mathcal T}_0(M))\), where \({\mathcal T}_0(M)\subset{\mathcal S}_0T^\ast M\otimes{\mathcal T}(M)\) is a subbundle defined by the natural split exact sequence \(0\to{\mathcal T}_0(M)\to{\mathcal S}_0T^\ast M\otimes{\mathcal T}M \to T^\ast M\to 0\). A projective connection is a class of projectively related affine connections \(\nabla\) and \({\mathcal L}\) assigns to a diffeomorphism \(f\) the \({\mathcal T}_0(M)\)-component of the difference tensor \((f^{-1})^\ast\nabla-\nabla\), which is independend of the choice of \(\nabla\). As a generalization of Gelfand-Fuchs cohomology, cohomology of Lie algebras of vector fields and of diffeomorphism groups with coefficients in various spaces of differential operators is considered.
Some appendices give five independent proofs of Sturms’ theorem, explain the language of symplectic and contact geometry, of connections, of homological algebra, cocycles on groups of diffeomorphisms, the Godbillon-Vey class, the Adler-Gelfand-Dickey bracket, and infinite dimensional Poisson geometry. The bibliography has 234 entries.
To sum up, this is an introduction to global projective differential geometry offering felicitous choice of topics, leading from classical projective differential geometry to current fields of reserach in mathematics and mathematical physics. The reader is guided from simple facts concerning curves and derivatives to more involved problems and methods through a world of inspiring ideas, delivering insights in deep relations. Historical comments as well as stimulating exercises occur frequently throughout the text, making it suitable for teachings.

MSC:

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53A20 Projective differential geometry
53D17 Poisson manifolds; Poisson groupoids and algebroids
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53B10 Projective connections
51N15 Projective analytic geometry
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