Abelian groups, module theory, and topology: proceedings in by Dikran Dikranjan, Luigi Salce

By Dikran Dikranjan, Luigi Salce

Encompasses a stimulating number of papers on abelian teams, commutative and noncommutative jewelry and their modules, and topological teams. Investigates at the moment renowned themes equivalent to Butler teams and nearly thoroughly decomposable teams.

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Extra info for Abelian groups, module theory, and topology: proceedings in honor of Adalberto Orsatti's 60th birthday

Example text

SECTIONAL CURVATURE OP HYPEKSURFACES. Consider a hypersurface W c V and let us formulate Gauss' teorema egregium which relates Kw(o) to Kv(o) for the planes o6 2\ o (F), w£W. For this we need the second fundamental form 11w restricted to o, where o is given the (Euclidean) metric inhereted from g on T,0(V). Now, every quadratic form on TR2 = (o, g\o} is characterized by its eigenvalues (which are the eigenvalues of the corresponding symmetric operator A on 1R-) and the product of these eigenvalues for the form JP¥ on o is denoted Dis(o), With this the Gauss formula reads Here as earlier the proof is algorithmic but the corollaries are quite nice.

G. \.. Then the small inward E-deformation of this piece-wise smooth hypersurface is again convex and piecewise smooth, where the deformed pieces may, unfortunately, have ||7I|| slightly greater than c. This increase of [|J7]|—*•<» which corresponds to the appearance of a focal point. But this can be prevented since the deformed hypersurface can be arbitrarily close approximated again by another piecewise smooth convex hypersurface having \\n\\ :£ c for all pieces. Thus- by sequentially applying small equidistant deformations followed by approximations we manage to keep in the category of piecewise smooth convex1 hypersurfaces for large inward deformations.

Such that w ^* 0 and W n V becomes a hypersurface W c U' c W passing through the origin. Then we define the form IIw of W cz: V at iv as that of W in Ifi" at 0, where the tangent spaces TX(W) and T0(W) are identified by the differential of the implied diffeomorphism U *-» V (sending W f\ U to W and TW(W) onto T,,{W')). A little thought shows this definition to be independent of the coordinates w, and with a minor extra effort one can see this form is the same as defined by the above {-(-).

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