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Special proper pointwise slant surfaces of a locally product Riemannian manifold

The structure of pointwise slant submanifolds in an almost product Riemannian manifold is investigated and the special proper pointwise slant surfaces of a locally product manifold are introduced. A relation involving the squared mean curvature and the Gauss curvature of pointwise slant surface of a locally product manifold is proved. | Turk J Math (2015) 39: 884 – 899 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1412-44 Research Article Special proper pointwise slant surfaces of a locally product Riemannian manifold 1,∗ ¨ ˇ ˙ 3 Mehmet GULBAHAR , Erol KILIC ¸ 2 , Semra SARAC ¸ OGLU C ¸ ELIK Department of Mathematics, Faculty of Science and Art, Siirt University, Siirt, Turkey 2 ˙ on¨ Department of Mathematics, Faculty of Science and Art, In¨ u University, Malatya, Turkey 3 Department of Mathematics, Faculty of Science, Bartın University, Bartın, Turkey 1 Received: 18.12.2014 • Accepted/Published Online: 07.06.2015 • Printed: 30.11.2015 Abstract: The structure of pointwise slant submanifolds in an almost product Riemannian manifold is investigated and the special proper pointwise slant surfaces of a locally product manifold are introduced. A relation involving the squared mean curvature and the Gauss curvature of pointwise slant surface of a locally product manifold is proved. Two examples of proper pointwise slant surfaces of a locally product manifold, one of which is special and the other one is not special, are given. Key words: Almost product Riemannian manifold, special slant surface, curvature 1. Introduction A slant surface M of a Kaehlerian manifold is called special slant if, with respect to some suitable adapted orthonormal frame {e1 , e2 , e3 , e4 } , the shape operator of the surface takes the following forms: ( A e3 = cλ 0 0 λ ) ( and Ae4 = 0 λ λ 0 ) , (1.1) where both c and λ are real numbers and {e1 , e2 } is an orthonormal basis on Tp M . The special slant surfaces were studied in complex space forms by Chen in [9] and [10]. He proved the following relation involving the squared mean curvature ∥H(p)∥2 and the Gauss curvature K at a point p of proper slant surface M in a f(4c): complex space form M ∥H(p)∥2 ≥ 2K(p) − 2(1 + cos2 θ)c, (1.2) where θ is the slant angle of the surface M . Furthermore, Chen showed