Abstract. We consider an immersed orientable hypersurface f : M → R n+1 of the Euclidean space (f an immersion), and observe that the tangent bundle T M of the hypersurface M is an immersed submanifold of the Euclidean space R 2n+2 . Then we show that in general the induced metric on T M is not a natural metric and obtain expressions for the horizontal and vertical lifts of the vector fields on M . We also study the special case in which the induced metric on T M becomes a natural metric and show that in this case the tangent bundle T M is trivial.
In this paper, we study 3-dimensional compact and connected trans-Sasakian manifolds and find necessary and sufficient conditions under which these manifolds are homothetic to Sasakian manifolds. First, four results in this paper deal with finding necessary and sufficient conditions on a compact and connected trans-Sasakian manifold to be homothetic to a compact and connected Sasakian manifold, and the fifth result deals with finding necessary and sufficient condition on a connected trans-Sasakian manifold to be homothetic to a connected Sasakian manifold. Finally, we find necessary and sufficient conditions on a compact and simply connected trans-Sasakian manifold to be homothetic to a compact and simply connected Einstein Sasakian manifold.
This work proposes a new numerical approach for dealing with fractional stochastic differential equations. In particular, a novel three-point fractional formula for approximating the Riemann–Liouville integrator is established, and then it is applied to generate approximate solutions for fractional stochastic differential equations. Such a formula is derived with the use of the generalized Taylor theorem coupled with a recent definition of the definite fractional integral. Our approach is compared with the approximate solution generated by the Euler–Maruyama method and the exact solution for the purpose of verifying our findings.
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