Intrinsic and extrinsic integral geometry

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AbstractAward: DMS-9972094Principal Investigator: Joseph H.G. FuWe propose to study the geometry of certain singular spaces,whose curvatures are naturally defined as measures. Thecurvatures under consideration are those arising in theSteiner-Weyl tube formula. For smooth spaces these quantitiesare curvature integrals. For singular spaces they are the totalvariations of certain signed measures which may be regarded asgeneralized curvatures. In all known cases these curvaturemeasures, which are a priori defined with respect to an embeddingof the singular space in Euclidean space, are in fact computablesolely from the intrinsic metric structure of the space. Ourgeneral goal is to understand and extend this phenomenon. Thisproblem was attacked independently by Alexandrov and by Federerin the 1950s. Alexandrov's approach was intrinsic but limited to2 dimensions, while Federer's theory dealt with embedded objectsof general dimension. Further progress will depend on thereconciliation of these two methods. We propose that thetechnical key is based on the use of the geometric measure theoryof currents. The starting point is the fact that Federer'stheory may be recast in the following terms: to certain singularsubspaces X in euclidean space there is an associated integralcurrent N(X) in the sphere bundle of the ambient space. Thiscurrent satisfies the Legendre condition (i.e. it annihilatesthe differential ideal generated by the canonical 1-form) and isdetermined by its relation with the Morse theory of X. Thecurvature measures of X are then computable in a canonical wayfrom N(X). Our proposal is based on recent work of the proposerusing the current theoretic framework, showing that Alexandrov'stheory may be understood in similar terms. This suggests thepossibility of generalizing the intrinsic approach to higherdimensions.In ordinary terms, this project aims to develop a mathematicallanguage adequate to describe geometrical forms we encounterevery day: for example, the crumplings of a sheet of paper or thesurface of a rock. Even though geometrical shapes typicallyfound in nature tend to be rough ("singular"), for technicalreasons geometry as it has developed historically restricts itsattention to smooth objects. Our goal is to be able to dealmathematically with such objects without imposing suchunrealistic assumptions. We expect that this work willeventually have a technological impact, in such areas as computervisualization and the study of buckling of sheets (as of a carfender during a collision).