Pure Maths Colloquium: The lightlike geometry of spacelike submanifolds in Minkowski space
13 December 2010 17:15 in CM221
Together with F.D.R. Fuster, we have investigated an extrinsic differential geometry on spacelike submanifolds of codimension two in Lorentz-Minkowski space. Examples of codimension two spacelike submanifolds are given by hypersurfaces in Euclidean space and Hyperbolic space.
Recently, we have discovered a new geometry on submanifolds in Hyperbolic space which is now called the horospherical geometry in Hyperbolic space. The horospherical geometry is quite different from the hyperbolic geometry in Hyperbolic space. However, it has similar properties to the Euclidean Differential Geometry. For example, the Gauss-Bonnet type theorem holds for the horospherical Gauss-Kronecker curvature. Moreover, the some part of geometric meanings for horo-tightness introduced by Cecil and Ryan have been recently clarified in the framework of the horospherical geometry.
The lightlike geometry for spacelike submanifolds in Minkowski space unifies the Euclidean Differential Geometry and Horospherical geometry in Hyperbolic space.
In this talk, we give the framework of the lightlike geometry on general spacelike submanifolds in Minkowski space. One of the results is the Chern-Lashof type theorem for the lightlike totally absolute curvature of spacelike submanifolds. Such a theorem naturally induces the notion of lightlike convexity and lightlike tightness etc.
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