HANDBOOK OF DIFFERENTIAL GEOMETRY
CONTENTS 1 SOME PROBLEMS ON FINSLER GEOMETRY 2 FOLIATIONS 3 SYMPLECTIC GEOMETRY 4 METRIC RIEMANNIAN GEOMETRY 5 CONTACT GEOMETRY 6 COMPLEX DIFFERENTIAL GEOMETRY 7 COMPENDIUM ON THE GEOMETRY OF LAGRANGE SPACES 8 CERTAIN ACTUAL TOPICS ON MODERN LORENTZIAN GEOMETRY CONTENTS OF VOLUME I 1 DIFFERENTIAL GEOMETRY OF WEBS 2 SPACES OF METRICS AND CURVATURE FUNCTIONALS 3 RIEMANNIAN SUBMANIFOLDS 4 EINSTEIN METRICS IN DIMENSION FOUR 5 THE ATIYAH–SINGER INDEX THEOREM 6 SURVEY OF ISOSPECTRAL MANIFOLDS 7 SUBMANIFOLDS WITH PARALLEL FUNDAMENTAL FORM 8 SPHERE THEOREMS 9 AF?NE DIFFERENTIAL GEOMETRY 10 A SURVEY ON ISOPARAMETRIC HYPERSURFACES AND THEIR GENERALIZATIONS 11 CURVES CHAPTER SOME PROBLEMS ON FINSLER GEOMETRY? CONTENTS 1 INTRODUCTION 2 PRELIMINARIES 21 THE HAMILTONIAN POINT OF VIEW 22 THE RIEMANNIAN POINT OF VIEW 23 ISOMETRIES AND ISOMETRIC EMBEDDINGS 24 ISOMETRIC SUBMERSIONS 3 VOLUME AND AREA IN FINSLER SPACES 4 UNIT SPHERES IN MINKOWSKI SPACES 5 SYMPLECTIC EQUIVALENCE OF FINSLER MANIFOLDS 51 EQUIVALENCE OF UNIT CO-DISC BUNDLES 52 EQUIVALENCE OF UNIT CO-SPHERE BUNDLES 53 MANIFOLDS OF GEODESICS 6 AROUND HILBERT’S FOURTH PROBLEM 7 CLOSED GEODESICS 8 DIFFERENTIAL INVARIANTS OF FINSLER SURFACES 81 CONVEX GEOMETRY AND THE INVARIANT I 82 THE INVARIANT J 83 CURVATURE OF FINSLER SURFACES 84 CARTAN’S STRUCTURE EQUATIONS 85 APPLICATIONS OF CARTAN’S STRUCTURE EQUATIONS REFERENCES HANDBOOK OF DIFFERENTIAL GEOMETRY ABSTRACT THIS CHAPTER IS AN UNORTHODOX SURVEY OF FINSLER GEOMETRY PRESENTING BOTH RESULTS AND OPEN PROBLEMS IT AIMS TO SHOW THAT RECENT PROGRESS IN CONVEX GEOMETRY, THE CALCULUS OF VARIATIONS, SYMPLECTIC GEOMETRY, AND INTEGRAL GEOMETRY CAN BE POWERFUL TOOLS IN THE STUDY OF FINSLER MANIFOLDS; AND THAT FINSLER GEOMETRY CAN PROVE USEFUL IN SOLVING SOME OF THE OPEN PROBLEMS IN THESE ?ELDS SOME PROBLEMS ON FINSLER GEOMETRY 1 INTRODUCTION FINSLER MANIFOLDS, MANIFOLDS WHOSE TANGENT SPACES CARRY A NORM THAT VARIES SMOOTHLY WITH THE BASE POINT, WERE BORN PREMATURELY IN 1854 TOGETHER WITH THEIR RIEMANNIAN COUNTERPARTS IN RIEMANN’S GROUND-BREAKING HABILITATIONSVORTRAG I SAY PREMATURELY BECAUSE IN