Approximation by a Composition of Apostol-Genocchi and P\v{a}lt\v{a}nea-Durrmeyer Operators


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Authors: N. S. MISHRA AND N. DEO

DOI: 10.46793/KgJMat2105.781S

Abstract:

In the present paper we consider some applications the Hopf maximum principle and its generalization to the classical theory of geodesic mappings. As a result, a series of classical theorems on geodesic mappings become consequences of our statements which we shall prove in the present paper.



Keywords:

Riemannian manifold, Einstein manifold, geodesic mapping, second order elliptic differential operator on symmetric tensors, Hopf maximum principle, vanishing theorems.



References:

[1]   I. A. Aleksandrova, J. Mikeš, S. E. Stepanov and I. I. Tsyganok, Liouville type theorems in the theory of mappings of complete Riemannian manifolds, J. Math. Sci. (N.Y.) 221 (2017), 737–744.

[2]   S. Bochner and K. Yano, Curvature and Betti Numbers, Princeton University Press, Princeton, 1953.

[3]   E. Calabi, An extension of E. Hopf’s maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958), 45–56.

[4]   X. Chen and Z. Shen, A comparison theorem on the Ricci curvature in projective geometry, Ann. Global Anal. Geom. 23 (2003), 141–155.

[5]   L. P. Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, New Jersey, 1949.

[6]   I. Hinterleitner, Geodesic mappings on compact Riemannian manifolds with conditions on sectional curvature, Publ. Inst. Math. (Beograd) (N.S.) 94(108) (2013), 125–130.

[7]   S. Kim, Volume and projective equivalence between Riemannian manifolds, Ann. Global Anal. Geom. 27 (2005), 47–52.

[8]   P. Li and R. Schoen, Lp and mean value properties of subharmonic functions on Riemannian manifolds, Acta Math. 153 (1984), 279–301.

[9]   J. Mikeš, E. Stepanova, A. Vanžurová and et al., Differential Geometry of Special Mappings, Palacký University Olomouc, Faculty of Science, Olomouc, 2015.

[10]   J. Mikeš, A. Vanžurová and I. Hinterleitner, Geodesic Mappings and some Generalizations, Palacký University Olomouc, Faculty of Science, Olomouc, 2009.

[11]   N. S. Sinyukov, Geodesic Mappings of Riemannian Spaces, Nauka, Moscow, 1979.

[12]   E. N. Sinyukova, On the geodesic mappings of some special Riemannian spaces, Math. Notes 30 (1981), 946–949.

[13]   S. E. Stepanov, I. I. Tsyganok and J. Mikeš, A Liouville type theorem on the projective mapping of a complete Riemannian manifold, Differentsial’naya Geom. Mnogoobraz. Figur (2017), 110–115.

[14]   S. T. Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), 659–670.