On the distance function to the cut locus of a submanifold in Finsler geometry [Elektronische Ressource] / vorgelegt von Niki Winter

On the distance function to the cut locusof a submanifold in Finsler geometryVon der Fakultät für Mathematik, Informatik und Naturwissenschaftender RWTH Aachen University zur Erlangung des akademischen Gradeseines Doktors der Naturwissenschaften genehmigte Dissertationvorgelegt vonDiplom–MathematikerNiki Winteraus HermeskeilBerichter: Univ.-Prof. Dr. Heiko von der MoselAOR Priv.-Doz. Dr. Alfred WagnerTag der mündlichen Prüfung: 19. November 2010Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek onlineverfügbar.iiAbstractIn the present thesis we study the distance function to the cut locus of a subman-ifold.WeexplainthebasicnotionsforagivenFinslermanifold (M;F )andasubmanifoldf fM. Roughly speaking, for a given pointx on the submanifoldM and any unit vector0fy which is normal toM we define the cut point as the first point on the geodesic0starting at x in direction y such that this geodesic fails to be minimising distance0 0ftoM for any point that lies beyond the cut point. The set of all cut points is calledfthe cut locus ofM and is denoted byCut . The distance function to the cut locus,fMi.e. the function that measures distance from x to the cut point, depends on (x ;y )0 0 0and is denoted by i . See Definition 2.15 for precise terminology.fMfWe prove that the distance function to the cut locus of a submanifoldM is locallyLipschitz continuous. For technical reasons, we have to presume the absence of con-jugate points, i.e.