PROJECTIVE DIFFERENTIAL GEOMETRY: FROM CLASSICAL TO MODERN APPROACHES

By ROBERT DAVIS

CONTENTS INTRODUCTION PROJECTIVE SPACE AND PROJECTIVE DUALITY DISCRETE INVARIANTS AND CON?GURATIONS INTRODUCING SCHWARZIAN DERIVATIVE FURTHER EXAMPLE OF DI?ERENTIAL INVARIANTS: PROJECTIVE CURVATURE SCHWARZIAN DERIVATIVE AS A COCYCLE OF DI?(RP1) VIRASORO ALGEBRA: THE COADJOINT REPRESENTATION GEOMETRY OF PROJECTIVE LINE INVARIANT DI?ERENTIAL OPERATORS ON RP1 CURVES IN RPN AND LINEAR DI?ERENTIAL OPERATORS HOMOTOPY CLASSES OF NON-DEGENERATE CURVES TWO DI?ERENTIAL INVARIANTS OF CURVES: PROJECTIVE CURVATURE AND CUBIC FORM PROJECTIVELY EQUIVARIANT SYMBOL CALCULUS ALGEBRA OF PROJECTIVE LINE AND COHOMOLOGY OF DI?(S1) TRANSVECTANTS FIRST COHOMOLOGY OF DI?(S1) WITH COE?CIENTS IN DI?ERENTIAL OPERATORS APPLICATION: GEOMETRY OF DI?ERENTIAL OPERATORS ON RP1 ALGEBRA OF TENSOR DENSITIES ON S1 EXTENSIONS OF VECT(S1) BY THE MODULES F?(S1) VERTICES OF PROJECTIVE CURVES CLASSIC 4-VERTEX AND 6-VERTEX THEOREMS GHYS’ THEOREM ON ZEROES OF THE SCHWARZIAN DERIVATIVE AND GEOMETRY OF LORENTZIAN CURVES BARNER THEOREM ON IN?ECTIONS OF PROJECTIVE CURVES APPLICATIONS OF STRICTLY CONVEX CURVES DISCRETIZATION: GEOMETRY OF POLYGONS, BACK TO CON?GURATIONS PROJECTIVE INVARIANTS OF SUBMANIFOLDS SURFACES IN RP3: DI?ERENTIAL INVARIANTS AND LOCAL GEOMETRY RELATIVE, A?NE AND PROJECTIVE DI?ERENTIAL GEOMETRY OF HYPER- SURFACES GEOMETRY OF RELATIVE NORMALS AND EXACT TRANSVERSE LINE ?ELDS COMPLETE INTEGRABILITY OF THE GEODESIC ?OW ON THE ELLIPSOID AND OF THE BILLIARD MAP INSIDE THE ELLIPSOID HILBERT’S 4-TH PROBLEM GLOBAL RESULTS ON SURFACES PROJECTIVE STRUCTURES ON SMOOTH MANIFOLDS DE?NITION, EXAMPLES AND MAIN PROPERTIES PROJECTIVE STRUCTURES IN TERMS OF DI?ERENTIAL FORMS TENSOR DENSITIES AND TWO INVARIANT DI?ERENTIAL OPERATORS PROJECTIVE STRUCTURES AND TENSOR DENSITIES MODULI SPACE OF PROJECTIVE STRUCTURES IN DIMENSION 2, BY V FOCK AND A GONCHAROV MULTI-DIMENSIONAL SCHWARZIAN DERIVATIVES AND DI?ERENTIAL OPERATORS PROJECTIVELY EQUIVARIANT SYMBOL CALCULUS IN ANY DIMENSION MULTI-DIMENSIONAL SCHWARZIAN AS A DI?ERENTIAL OPERATOR ?(M) FOR AN ARBITRARY MANIFOLD POISSON ALGEBRA OF TENSOR DENSITIES ON A CONTACT MANIFOLD LAGRANGE SCHWARZIAN DERIVATIVE CHAPTER PROJECTIVE SPACE AND PROJECTIVE DUALITY GIVEN A VECTOR SPACE V , THE ASSOCIATED PROJECTIVE SPACE, P(V ), CONSISTS OF ONE-DIMENSIONAL SUBSPACES OF V IF V = RN+1 THEN P(V ) IS DENOTED BY RPN THE PROJECTIVIZATION, P(U), OF A SUBSPACE U ?V IS CALLED A PROJECTIVE SUBSPACE OF P(V ) CHAPTER 1 INTRODUCTION THE DUAL PROJECTIVE SPACE P(V )?IS THE PROJECTIVIZATION OF THE DUAL VEC- TOR SPACE V ? PROJECTIVE DUALITY IS A CORRESPONDENCE BETWEEN PROJECTIVE SUBSPACES OF P(V ) AND P(V )?, THE RESPECTIVE LINEAR SUBSPACES OF V AND V ? ARE ANNULATORS OF EACH OTHER NOTE THAT PROJECTIVE DUALITY REVERSES THE INCI- DENCE RELATION NATURAL LOCAL COORDINATES ON RPN COME FROM THE VECTOR SPACE RN+1 IF X0, X1, , XN ARE LINEAR COORDINATES IN RN+1, THEN YI = XI/X0 ARE CALLED A?NE COORDINATES ON RPN; THESE COORDINATES ARE DE?NED IN THE CHART X0 ?= 0 LIKEWISE, ONE DE?NES A?NE CHARTS XI ?= 0 THE TRANSITION FUNCTIONS BETWEEN TWO A?NE COORDINATE SYSTEMS ARE FRACTIONAL-LINEAR PROJECTIVELY DUAL CURVES IN DIMENSION THE PROJECTIVE DUALITY EXTENDS TO CURVES A SMOOTH CURVE ? IN RP2 DETER- MINES A 1-PARAMETER FAMILY OF ITS TANGENT LINES EACH OF THESE LINES GIVES A POINT IN THE DUAL PLANE RP2?AND WE OBTAIN A NEW CURVE ??IN RP2?, CALLED THE DUAL CURVE IN A GENERIC POINT OF ?, THE DUAL CURVE IS SMOOTH POINTS IN WHICH ??HAS SINGULARITIES CORRESPOND TO IN?ECTION OF ? IN GENERIC POINTS, ? HAS ORDER CONTACT WITH ITS TANGENT LINE; IN?ECTION POINTS ARE THOSE POINTS WHERE THE ORDER OF CONTACT IS HIGHER ? ?? FIGURE 1 1: DUALITY BETWEEN AN IN?ECTION AND A CUSP EXERCISE 1 1 1 A) TWO PARABOLAS, GIVEN IN A?NE COORDINATES BY Y = X? AND Y = X?, ARE DUAL FOR 1/? + 1/? = 1 B) THE CURVES IN ?GURE 1 2 ARE DUAL TO EACH OTHER A FUNDAMENTAL FACT IS THAT (??)?= ? WHICH JUSTI?ES THE TERMINOLOGY (A PROOF GIVEN IN THE NEXT SUBSECTION) AS A CONSEQUENCE, ONE HAS AN ALTERNATIVE DE?NITION OF THE DUAL CURVE EVERY POINT OF ? DETERMINES A LINE IN THE DUAL PLANE, AND THE ENVELOPE OF THESE LINES IS ?? TWO REMARKS ARE IN ORDER THE DE?NITION OF THE DUAL CURVE EXTENDS TO CURVES WITH CUSPS, PROVIDED THE TANGENT LINE IS DE?NED AT EVERY POINT AND PROJECTIVE DIFFERENTIAL GEOMETRY OLD AND NEW: FROM SCHWARZIAN 1 1 PROJECTIVE SPACE AND PROJECTIVE DUALITY 2 I = N T+ T- I N FIGURE 1 3: INVARIANTS OF PLANE CURVES HINT ORIENT ? AND LET ?(X) BE THE POSITIVE TANGENT RAY AT X ?? CONSIDER THE NUMBER OF INTERSECTION POINTS OF ?(X) WITH ? AND INVESTIGATE HOW THIS NUMBER CHANGES AS X TRAVERSES ? DO THE SAME WITH THE NEGATIVE TANGENT RAY PROJECTIVE CURVES IN HIGHER DIMENSIONS CONSIDER A GENERIC SMOOTH PARAMETERIZED CURVE ?(T) IN RPN AND ITS GENERIC POINT ?(0) CONSTRUCT A ?AG OF SUBSPACES AS FOLLOWS FIX AN A?NE COOR- PROJECTIVE DIFFERENTIAL GEOMETRY OLD AND NEW: FROM SCHWARZIAN