Yazar "Deshmukh, Sharief" seçeneğine göre listele

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    Frenet Curves in Euclidean 4-Space
    (Int Electronic Journal Geometry, 2017) Deshmukh, Sharief; Al-Dayel, Ibrahim; Ilarslan, Kazim
    In this paper, we study rectifying curves arising through the dilation of unit speed curves on the unit sphere S-3 and conclude that arcs of great circles on S-3 do not dilate to rectifying curves, which develope previously obtained results for rectifying curves in Eucidean spaces. This fact allows us to prove that there exists an associated rectifying curve for each Frenet curve in the Euclidean space E-4 and a result of the fact rectifying curves in the Euclidean space E-4 are ample, indeed as an appication, we present an ordinary differential equation satisfied by the distance function of a Frenet curve in E-4 which alows us to characterize the spherical curves and rectifying curves in E-4. Furthermore, we study ccr-curves in the Euclidean space E-4 which are generalizations of helices in E-3 and show that the property of a helix that its tangent vector field makes a constant angel with a fixed vector (axis of helix) does not go through for a ccr-curve.
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    Spheres and Euclidean Spaces Via Concircular Vector Fields
    (Springer Basel Ag, 2021) Deshmukh, Sharief; İlarslan, Kazım; Alsodais, Hana; De, Uday Chand
    In this paper, we exhibit that non-trivial concircular vector fields play an important role in characterizing spheres, as well as Euclidean spaces. Given a non-trivial concircular vector field xi on a connected Riemannian manifold (M, g), two smooth functions s and. called potential function and connecting function are naturally associated to xi. We use non-trivial concircular vector fields on n-dimensional compact Riemannian manifolds to find four different characterizations of spheres S-n (c). In particular, we prove an interesting result namely an n-dimensional compact Riemannian manifold (M, g) that admits a non-trivial concircular vector field xi such that the Ricci operator is invariant under the flow of xi, if and only if, (M, g) is isometric to a sphere Sn (c). Similarly, we find two characterizations of Euclidean spaces E-n. In particular, we show that an n-dimensional complete and connected Riemannian manifold (M, g) admits a non-trivial concircular vector field xi that annihilates the Ricci operator, if and only if, (M, g) is isometric to the Euclidean space E-n.