Abstract
According to Tashiro–Obata, on a Riemannian manifold (M, g) with its Ricci curvature bounded positively from below, the first eigenvalue of the Laplacian on functions satisfies a simple inequality in terms of the scalar curvature, and equality characterizes the Riemannian sphere. We discuss a similar inequality for a certain elliptic operator on a manifold with conjugate connections. As application we characterize hyperellipsoids in Blaschke’s unimodular-affine hypersurface theory.
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Berger, M., Gauduchon, P., Mazet, E.: Le spectre d’une variété Riemannienne, Springer Lecture Notes, vol. 194 (1971)
Blaschke, W.: Vorlesungen über Differentialgeometrie. II. Affine Differential-geometrie. Springer, Berlin (1923)
Gardner R.B., Kriele M., Simon U.: Generalized spherical functions on projectively flat manifolds. Results Math. 27, 41–50 (1995)
Myers S.B.: Riemannian manifolds in the large. Duke Math. J. 1, 39–49 (1935)
Opozda, B.: Bochner’s Technique for Statistical Manifolds. JU Krakow (2013, manuscript)
Schneider R.: Zur affinen Differentialgeometrie im Großen. I. Math. Z. 101, 375–406 (1967)
Sheng, L., Li, A.M., Simon, U.: Complete Blaschke hypersurfaces with negative affine mean curvature. Ann. Glob. Anal. Geom. (2014, to appear)
Simon, U., Schwenk-Schellschmidt, A., Viesel, H.: Introduction to the Affine Hypersurface Theory, Lecture Notes, Science University Tokyo (1991)
Simon, U.: Affine Hypersurface Theory Revisited: Gauge Invariant Structures (English version), Russian Mathematics (Izv. vuz) vol. 48, pp. 48–73 (2004)
Wiehe, M.: Deformations in Affine Hypersurface Theory. Doctoral Thesis, FB Mathematik TU Berlin (1998)
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Simon, U. The first eigenvalue of Laplace-type elliptic operators induced by conjugate connections. J. Geom. 106, 313–320 (2015). https://doi.org/10.1007/s00022-014-0250-2
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DOI: https://doi.org/10.1007/s00022-014-0250-2