THE GEOMETRY OF TOTAL CURVATURE ON COMPLETE OPEN SURFACES

By JOSHUA GARCIA

CONTENTS RIEMANNIAN GEOMETRY THE RIEMANNIAN METRIC GEODESICS THE RIEMANNIAN CURVATURE TENSOR THE SECOND FUNDAMENTAL FORM THE SECOND VARIATION FORMULA AND JACOBI ?ELDS COMPLETE RIEMANNIAN MANIFOLDS THE SHORT-CUT PRINCIPLE THE GAUSS–BONNET THEOREM THE CLASSICAL RESULTS OF COHN-VOSSEN AND HUBER THE TOTAL CURVATURE OF COMPLETE OPEN SURFACES THE CLASSICAL THEOREMS OF COHN-VOSSEN AND HUBER SPECIAL PROPERTIES OF GEODESICS ON RIEMANNIAN PLANES THE IDEAL BOUNDARY THE CURVATURE AT IN?NITY PARALLELISM AND PSEUDO-DISTANCE BETWEEN CURVES RIEMANNIAN HALF-CYLINDERS AND THEIR UNIVERSAL COVERINGS THE IDEAL BOUNDARY AND ITS TOPOLOGICAL STRUCTURE THE STRUCTURE OF THE TITS METRIC D? TRIANGLE COMPARISON CONVERGENCE TO THE LIMIT CONE THE BEHAVIOR OF BUSEMANN FUNCTIONS THE CUT LOCI OF COMPLETE OPEN SURFACES PRELIMINARIES THE TOPOLOGICAL STRUCTURE OF A CUT LOCUS ABSOLUTE CONTINUITY OF THE DISTANCE FUNCTION OF THE CUT LOCUS THE STRUCTURE OF GEODESIC CIRCLES ISOPERIMETRIC INEQUALITIES THE STRUCTURES OF S(C, T) AND THE CUT LOCUS OF C THE CASE WHERE M IS ?NITELY CONNECTED THE CASE WHERE M IS IN?NITELY CONNECTED MASS OF RAYS PRELIMINARIES; THE MASS OF RAYS EMANATING FROM A ?XED POINT ASYMPTOTIC BEHAVIOR OF THE MASS OF RAYS THE POLES AND CUT LOCI OF A SURFACE OF REVOLUTION PROPERTIES OF GEODESICS JACOBI ?ELDS THE CUT LOCI OF A VON MANGOLDT SURFACE THE BEHAVIOR OF GEODESICS THE SHAPE OF PLANE CURVES MAIN THEOREMS AND EXAMPLES THE SEMI-REGULARITY OF GEODESICS ALMOST-REGULARITY OF GEODESICS; ESTIMATE OF INDEX THE ROTATION NUMBERS OF PROPER COMPLETE GEODESICS THE EXISTENCE OF COMPLETE GEODESICS ARBITRARILY CLOSE TO IN?NITY REFERENCES