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    Anti-invariant Riemannian submersions from Sasakian manifolds with totally umbilical fibers
    (World Scientific Publ Co Pte Ltd, 2021) Koprulu, Gizem; Sahin, Bayram
    The purpose of this paper is to study anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds such that characteristic vector field is vertical or horizontal vector field. We first show that any anti-invariant Riemannian submersions from Sasakian manifold is not a Riemannian submersion with totally umbilical fiber. Then we introduce anti-invariant Riemannian submersions from Sasakian manifolds with totally contact umbilical fibers. We investigate the totally contact geodesicity of fibers of such submersions. Moreover, under this condition, we investigate Ricci curvature of anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds.
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    Applications of Riemannian Submersions
    (Academic Press Ltd-Elsevier Science Ltd, 2017) Sahin, Bayram
    In this chapter, we give two applications of Riemannian submersions and it consists of two sections. In the first section, we give an application of Riemannian submersions in robotic theory [12]. To this aim, we first define Lie groups and give basic properties of such manifolds, then we give applications of Riemannian submersions in redundant robotic theory. Applications of Riemannian submersions in Kaluza-Klein theory have been given in detail in the book [111] and papers [112] and [40]; therefore, in the second section of this chapter, we give brief information for the Kaluza-Klein theory in terms of principal fiber bundles and Riemannian submersions.
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    Basic Geometric Structures on Manifolds
    (Academic Press Ltd-Elsevier Science Ltd, 2017) Sahin, Bayram
    In this chapter, we give brief information about geometric structures which will be used in the following chapters. In addition to the basic concepts, some geometric notions such as symmetry conditions, parallelity conditions, new product structures on manifolds, generalized Einstein manifolds, and biharmonic maps introduced very recently will be presented in this chapter. The chapter consists of six sections. In the first section, the main notions and theorems of Riemannian geometry are reviewed, such as the Riemannian manifold, Riemannian metric, Riemannian connection, curvatures (Riemannian, sectional Ricci, Scalar), derivatives (exterior, covariant, inner), operators (Hessian, divergence, Laplacian), Einstein manifolds and their generalizations, symmetry conditions, and very brief notions from integration. In the second section, we look at vector bundles and related notions. In the third section, we recall basic notions from submanifold theory and distributions on manifolds. In the fourth section, we mention Riemannian submersions, O'Neill's tensor fields, and curvature relations of Riemannian submersions. In this section, we also recall the notion of horizontally weakly conformal maps, which will be a crucial tool for a characterization of harmonic maps. In the fifth section, we study various product structures including warped product, twisted product, oblique warped product, and convolution product on Riemannian manifolds. We also provide connection relations and curvature expressions of each case. In the last section, we give a general setting for a map between manifolds such as a vector field along a map, a connection along a map, a curvature tensor field along a map, a second fundamental form of a map, a tension field of a map. We also recall harmonic maps and biharmonic maps in detail.
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    BETA BEZIER CURVES
    (Ministry Communications & High Technologies Republic Azerbaijan, 2019) Levent, Akin; Sahin, Bayram
    In this paper, we investigate geometric properties of such curves. We construct a new curve by using curves based on new basis with beta functions satisfying various continuity condition and check shape properties. As an interesting property of such curves, we show that the curves based on new basis with beta functions do not have loops which is a useful property for CAGD systems. Since Pythagoren hodograph curves are also important in various CAD applications, we find certain conditions for Pythagorean Hodograph for the curves based on new basis with beta functions.
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    BI-TANGENT QUATERNION KAEHLER MANIFOLDS
    (Univ Politehnica Bucharest, Sci Bull, 2024) Poyraz, Deniz; Sahin, Bayram
    Complex structures and tangent structures are well known. Almost semi-quaternion manifolds formed by the combination of these two structures have been also studied. In this paper, firstly, the existence of a structure similar to the quaternion Kaehler manifold for an almost semi-quaternion manifold is investigated. For this purpose, the necessary conditions are obtained for the covariant derivative of each cross-section to remain in the vector bundle formed by these cross -sections. After this stage, the interactions of these cross -sections and the curvature tensor field are investigated. In this direction, new relations were found. Finally, the flatness of the manifolds with this type of structure in case of constant curvature is examined.
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    Certain curves along Riemannian submersions
    (Univ Nis, Fac Sci Math, 2023) Tukel, Gozde Ozkan; Sahin, Bayram; Turhan, Tunahan
    In this paper, when a given curve on the total manifold of a Riemannian submersion is transferred to the base manifold, the character of the corresponding curve is examined. First, the case of a Frenet curve on the total manifold being a Frenet curve on the base manifold along a Riemannian submersion is investigated. Then, the condition that a circle on the total manifold (respectively a helix) is a circle (respectively, a helix) or a geodesic on the base manifold along a Riemannian submersion is obtained. We also investigate the curvatures of the original curve on the total manifold and the corresponding curve on the base manifold in terms of Riemannian submersions.
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    Certain Lie algebraic structures on Riemannian manifolds with semi-symmetric non-metric connection
    (Univ Nis, Fac Sci Math, 2023) Sahin, Fulya; Sahin, Bayram
    As a natural consequence of the Levi-Civita connection on a Riemannian manifold, there is a Lie algebra structure on a Riemannian manifold. Lie Algebras and Lie Groups are the mathematical structure of continuous symmetries in physics. In this paper, semi-symmetric non-metric connection is considered instead of Levi-Civita connection of Riemann manifold, and accordingly the existence of algebraic structures is investigated. First, it is shown that there is not always a Lie algebra structure on a Riemannian manifold with a semi-symmetric non-metric connection. Then, necessary and sufficient conditions for Lie admissible algebra, pre-Lie algebra and post Lie algebra on a Riemann manifold with semi-symmetric non-metric connection are obtained depending on geometric terms. In addition, the cases of the Riemannian manifold with such algebraic structures according to the semi-symmetric non-metric connection being Einstein manifold and being flat manifold have been also investigated.
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    Characterizations of pseudo-umbilical submanifolds of Kaehler manifolds by means of specific types
    (Springer Basel Ag, 2022) Sahin, Bayram
    As is well known, pseudo-umbilical submanifolds are the broad class of submanifolds that include totally umbilical submanifolds. Compared to totally umbilical submanifolds, studies on pseudo-umbilical submanifolds of Kaehler manifolds are quite limited. In this paper, necessary and sufficient conditions for pseudo-umbilical submanifolds of a Kaehler manifold to be a holomorphic submanifold, a totally real submanifold or a CR-submanifold are found in terms of projections defined on the submanifold.
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    Chen's first inequality for Riemannian maps
    (Polish Acad Sciences Inst Mathematics-Impan, 2016) Sahin, Bayram
    We obtain a basic Chen inequality for Riemannian maps between Riemannian manifolds.
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    Chen-Ricci inequalities for Riemannian maps and their applications
    (Amer Mathematical Soc, 2022) Lee, Jae Won; Lee, Chul Woo; Sahin, Bayram; Vilcu, Gabriel-Eduard
    Riemannian maps between Riemannian manifolds, originally introduced by A.E. Fischer in [Contemp. Math. 132 (1992), 331-366], provide an excellent tool for comparing the geometric structures of the source and target manifolds. Isometric immersions and Riemannian submersions are particular examples of such maps. In this work, we first prove a geometric inequality for Riemannian maps having a real space form as a target manifold. Applying it to the particular case of Riemannian submanifolds, we recover a classical result, obtained by B.-Y. Chen in [Glasgow Math. J. 41 (1999), 33-41], which nowadays is known as the Chen-Ricci inequality. Moreover, we extend this inequality in case of Riemannian maps with a complex space form as a target manifold. We also improve this inequality when the Riemannian map is Lagrangian. Applying it to Riemannian submanifolds, we recover the improved Chen-Ricci inequality for Lagrangian submanifolds in a complex space form, that is a basic inequality obtained by S. Deng in [Int. Electron. Electron. J. Geom. 2 (2009), 39-45] as an improvement of a geometric inequality stated by B.-Y. Chen in [Arch. Math. (Basel) 74 (2000), 154-160].
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    CIRCLES ALONG A RIEMANNIAN MAP AND CLAIRAUT RIEMANNIAN MAPS
    (Korean Mathematical Soc, 2017) Sahin, Bayram
    We first extend Yano-Nomizu's theorem, which characterizes extrinsic spheres in a Riemannian manifold, for Riemannian maps. Then we introduce Clairaut Riemannian maps, give an example and obtain necessary and sufficient conditions for a Riemannian map to be Clairaut type.
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    CONFORMAL ANTI-INVARIANT RIEMANNIAN MAPS TO KAHLER MANIFOLDS
    (Univ Politehnica Bucharest, Sci Bull, 2018) Akyol, Mehmet Akif; Sahin, Bayram
    We introduce conformal anti-invariant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds and show that they include both anti-invariant submanifolds and anti-invariant Riemannian maps. We give non-trivial examples, investigate the geometry of certain distributions and obtain decomposition theorems for the base manifold. The harmonicity and totally geodesicity of conformal anti-invariant Riemannian maps are also obtained. Moreover, we study weakly umbilical conformal Riemannian maps and obtain a classification theorem for umbilical conformal anti-invariant Riemannian maps.
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    Conformal Riemannian maps from almost Hermitian manifolds
    (Scientific Technical Research Council Turkey-Tubitak, 2018) Sahin, Bayram; Yanan, Sener
    Conformal Riemannian maps from almost Hermitian manifolds to Riemannian manifolds, namely conformal invariant Riemannian maps, holomorphic conformal Riemannian maps, and conformal antiinvariant Riemannian maps, are introduced. We mainly focus on conformal antiinvariant Riemannian maps from Kaehlerian manifolds. We give proper examples of conformal antiinvariant Riemannian maps, obtain the integrability of certain distributions, and investigate the geometry of leaves of these distributions. We also obtain various conditions for such maps to be horizontally homothetic maps.
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    Conformal Semi-invariant Riemannian Maps from Almost Hermitian Manifolds
    (Univ Nis, Fac Sci Math, 2019) Sahin, Bayram; Yanan, Sener
    Conformal semi-invariant Riemannian maps from Kaehler manifolds to Riemannian manifolds are introduced. We give examples, study the geometry of leaves of certain distributions and investigate certain conditions for such maps to be horizontally homothetic. Morever, we introduce special pluriharmonic maps and obtain characterizations.
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    CONFORMAL SEMI-INVARIANT RIEMANNIAN MAPS TO KAHLER MANIFOLDS
    (Union Matematica Argentina, 2019) Akyol, Mehmet Akif; Sahin, Bayram
    As a generalization of CR-submanifolds and semi-invariant Riemannian maps, we introduce conformal semi-invariant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds. We give non-trivial examples, investigate the geometry of foliations, and obtain decomposition theorems by using the existence of conformal Riemannian maps. We also investigate the harmonicity of such maps and find necessary and sufficient conditions for conformal anti-invariant Riemannian maps to be totally geodesic.
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    Conformal Slant Riemannian Maps to Kahler Manifolds
    (Tokyo Journal Mathematics Editorial Office Acad Center, 2019) Akyol, Mehmet Akif; Sahin, Bayram
    As a generalization of slant submanifolds and slant Riemannian maps, we introduce conformal slant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds. We give non-trivial examples, investigate the geometry of foliations and obtain decomposition theorems by using the existence of conformal Riemannian maps. Moreover, we also investigate the harmonicity of such maps and find necessary and sufficient conditions for conformal slant Riemannian maps to be totally geodesic.
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    Conformal slant submersions
    (Hacettepe Univ, Fac Sci, 2019) Akyol, Mehmet Akif; Sahin, Bayram
    As a generalization of conformal holomorphic submersions and conformal anti-invariant submersions, we introduce a new conformal submersion from almost Hermitian manifolds onto Riemannian manifolds, namely conformal slant submersions. We give examples and find necessary and sufficient conditions for such maps to be harmonic morphism. We also investigate the geometry of foliations which are arisen from the definition of a conformal submersion and obtain a decomposition theorem on the total space of a conformal slant submersion. Moreover, we find necessary and sufficient conditions of a conformal slant submersion to be totally geodesic.
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    CUBIC BEZIER-LIKE TRANSITION CURVES WITH NEW BASIS FUNCTIONS
    (Inst Mathematics & Mechanics, Natl Acad Sciences Azerbaijan, 2018) Levent, Akin; Sahin, Bayram
    A cubic trigonometric Bezier-like curve similar to the cubic Bezier curve, with a shape parameter, is presented in this work. With the suitable shape parameter, the cubic trigonometric Bezier-like curves can be made close to the cubic Bezier curves. Then we investigate the condition of smooth connection between non-adjacent cubic trigonometric Bezier-like curves.
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    Generalized almost para-contact manifolds
    (World Scientific Publ Co Pte Ltd, 2017) Sahin, Bayram; Sahin, Fulya
    In this paper, we study generalized almost para-contact manifolds and obtain normality conditions in terms of classical tensor fields. We show that such manifolds naturally carry certain Lie bialgebroid/quasi-Lie algebroid structures on them and we relate these new generalized manifolds with classical almost para-contact manifolds. The paper contains several examples and a short review for relations between generalized geometry and string theory.
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    Golden maps between golden Riemannian manifolds and constancy of certain maps(vol 19, pg 333, 2014)
    (Univ Osijek, Dept Mathematics, 2017) Sahin, Bayram; Akyol, Mehmet Akif