## An optimal inequality and an extremal class of graph hypersurfaces in affine geometry.(English)Zbl 1076.53010

The author considers a locally strongly convex graph with special relative normalization and the local geometry induced from it. The integrability conditions for a relative geometry imply the relative theorema egregium [see e.g. p. 79 in: U. Simon, A. Schwenk-Schellschmidt, H. Viesel, “Introduction to the affine differential geometry of hypersurfaces”, Lecture Notes Science University Tokyo (SUT), Dept. of Mathematics (1992; Zbl 0780.53002)]: $\text{trace}_h \text{Ric} = \|K\|^2 + (n-1)\operatorname{trace} S - n^2 \|T\|^2.$ Here $$h$$ denotes the relative metric, Ric its Ricci tensor, $$S$$ the induced relative Weingarten operator, $$K:= \nabla - \nabla(h)$$ the difference tensor between the induced connection $$\nabla$$ and the Levi-Civita connection $$\nabla(h)$$ of $$h$$, and $$nT$$ the trace of $$K$$, the so called Chebyshev form. The author states the following inequality as Theorem 1: The relative geometry of a graph hypersurface with a constant normalization ($$S=0$$) satisfies $\text{trace}_h \text{Ric} \geq \frac{n^2 (1-n)}{n+2} \|T\|^2.$ In Theorem 2 he gives a classification of all hypersurfaces satisfying equality in Theorem 1, using some explicit coordinate representations.
Reviewer’s remarks. 1. One can simplify the proof of Theorem 1, inserting the traceless part $$\widetilde {K}$$ of $$K$$ into the theorema egregium. Equality in Theorem 1 holds if and only if $$\widetilde {K}= 0$$, that means if the hypersurface is a quadric [see section 7.1 in the lecture notes cited above].
2. Concerning the foregoing remark, the reviewer contacted the author; using Mathematica, the author could verify that all hypersurfaces in his classification are quadrics, and he could give coordinate representations in terms of quadratic polynomials.
3. The author pointed out typing errors. In formula type (IV), Theorem 2, $$\sin$$ and $$\cos$$ shall read $$\sinh$$ and $$\cosh$$.
Reviewer: Udo Simon (Berlin)

### MSC:

 53A15 Affine differential geometry

Zbl 0780.53002

Mathematica
Full Text:

### References:

 [1] Chen, B.-Y.: Geometry of Submanifolds. Pure and Applied Mathematics, no.,22, Marcel Dekker, New York (1973). · Zbl 0262.53036 [2] Chen, B.-Y.: An optimal inequality and extremal classes of affine spheres in centroaffine geometry. (To appear in Geom. Dedicata.) · Zbl 1077.53012 · doi:10.1007/s10711-004-4199-4 [3] Hiepko, S.: Eine innere Kennzeichnung der verzerrten Produkts. Math. Ann., 241 , 209-215 (1979). · Zbl 0387.53014 · doi:10.1007/BF01421206 [4] Nomizu, K., and Pinkall, U.: On the geometry of affine immersions. Math. Z., 195 , 165-178 (1987). · Zbl 0629.53012 · doi:10.1007/BF01166455 [5] Nomizu, K., and Sasaki, T.: Affine Differential Geometry. Geometry of Affine Immersions. Cambridge Tracts in Math., no.,111, Cambridge University Press, Cambridge (1994). · Zbl 0834.53002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.