By Thomas E. Cecil
This exposition offers the state-of-the paintings at the differential geometry of hypersurfaces in actual, complicated, and quaternionic house kinds. precise emphasis is put on isoparametric and Dupin hypersurfaces in genuine area types in addition to Hopf hypersurfaces in complicated house kinds. The ebook is obtainable to a reader who has accomplished a one-year graduate direction in differential geometry. The textual content, together with open difficulties and an in depth checklist of references, is a superb source for researchers during this area.
Geometry of Hypersurfaces starts with the elemental concept of submanifolds in genuine house types. issues contain form operators, vital curvatures and foliations, tubes and parallel hypersurfaces, curvature spheres and focal submanifolds. the focal point then turns to the speculation of isoparametric hypersurfaces in spheres. very important examples and class effects are given, together with the development of isoparametric hypersurfaces in accordance with representations of Clifford algebras. An in-depth therapy of Dupin hypersurfaces follows with effects which are proved within the context of Lie sphere geometry in addition to those who are bought utilizing typical tools of submanifold idea. subsequent comes a radical remedy of the speculation of genuine hypersurfaces in advanced area kinds. A critical concentration is a whole evidence of the type of Hopf hypersurfaces with consistent vital curvatures because of Kimura and Berndt. The booklet concludes with the fundamental thought of actual hypersurfaces in quaternionic house types, together with statements of the most important class effects and instructions for additional research.
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Hgrad h; i/ be the corresponding principal curvature of multiplicity m D 1 of the hypersurface . ı f / W M ! RnC1 (respectively, DnC1 ), where is stereographic projection. Then X D 0 at x 2 M if and only if X D 0 at x. Proof. We will do the proof for a hypersurface in SnC1 , and the proof in H nC1 is quite similar. This is a local calculation, so we will consider M as an embedded hypersurface in SnC1 and suppress the mention of the embedding f . We use stereographic projection W SnC1 fPg ! x/ D1 hx; Pi.
For a unit vector Á 2 Tx? V, let BÁ denote the shape operator of V corresponding to Á. x/, and thus BÁ D I. Next let Á 2 T ? x/ be a unit length principal vector of A with corresponding principal curvature , so that AÁ D Á for ¤ . Extend Á to a vector field Y 2 T ? on a neighborhood W of x. Then there exists a unique vector field Z 2 T ? 57) for some smooth function on W. This is possible since T ? is invariant under A, even though the eigenvalues of A need not be smooth. We now find the shape operator BÁ .
Grad h; // is a smooth principal curvature function of multiplicity m on respective principal distributions of and coincide on M. 7 (Stereographic projection and inversions in spheres). 6 to the case of hypersurfaces in real space forms by considering the conformal transformation given by stereographic projection from Sk or H k into Rk , for any positive integer k. 44) where h ; i is the Euclidean inner product on RkC1 . Then stereographic projection with pole P is the map W Sk fPg ! Rk defined geometrically as follows.