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

Download PDF

Authors: N. S. MISHRA AND N. DEO

DOI: 10.46793/KgJMat2105.781S


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.


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


[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.