Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

We consider an immersed orientable hypersurface f : M ! R n+1 of the Euclidean space (f an immersion), and observe that the tan- gent bundle TM 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 TM 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 TM becomes a natural metric and show that in this case the tangent bundle TM is trivial.

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Tangent Bundle of the Hypersurfaces in a Euclidean Space

The works presented in this thesis explore the non-linearity and non-stationarity aspects of the hydroclimate system in France and the Euro-Atlantique area. In part I, «Spatio-temporal scales of hydroclimate variability in France», classifications of precipitation, temperature and discharge time series of 152 watersheds were established, based on non-linear and non-stationary characteristics. In particular, we based our classifications on spectral contents and cross-scale, phase-phase, phase-amplitude interactions. Results show clear homogeneous regions for each variable, but with complex spectral content and cross-scale interactions. Part II, « Spatiotemporal of local and large-scale climate variability», we studied how the watershed modulation altered the spectral correlation between local climate (precipitation, temperature) and discharge. We then studied the large-scale atmospheric patterns linked to discharge, with a combined spatial and temporal spectral analysis. Results show than temperature is the main driver of discharge at the annual scale, but precipitation is the main driver for other time scales. The large scale atmospheric patterns linked to discharge were shown to be time scale dependent and heavily modulated from West to east, with interacting atmospheric components shifting the main time scales of spectral correlation with discharge. Finally, Part III « North Atlantic atmospheric circulation dynamics», introduces a new conjecture linking the geometrical theory of the extremal length (a conformal invariant) and conservative atomspheric systems, such as the North Atlantic atmospheric circulation. We have shown that the time invariance of the extremal length implies conservation in mechanical energy within the atmospheric system, and that the increase/decrease of invariance induces mechanical energy losses/gains. The application of the theory to the North Atlantic atmospheric circulation, shows that depending on the time scale, the dynamics are vastly different, ranging for asymptotically stable to unstable, and that the atmospheric circulation for the last 40 years has seen large changes in its dynamics. Through this thesis, we showed the importance of time scale dependence, interactions between them, and more generally, of non-linearity and non-stationarity. We encourage hydrologist and climate scientists to adopt a combined physics-statistical approach in order to better grasp the complexity of hydroclimate systems.

Non-linear and non-stationary hydroclimate variability in France and the Euro-Atlantic area

We discuss the fundamental theoretical framework together with numerous results obtained by the authors and colleagues over an extended period of investigation on the Information Geometric Approach to Chaos (IGAC).

Information Geometric Measures of Complexity with Applications to Classical and Quantum Physical Settings

An isometric immersion of a Riemannian manifold M into a Riemannian manifold N gives rise in a natural way to the immersion of the tangent bundle TM into the tangent bundle TN with a non-degenerate g- natural metric G.

On a class of submanifolds in tangent bundle with g- natural metric

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.

Some Conditions on Trans-Sasakian Manifolds to Be Homothetic to Sasakian Manifolds

In this article, it has been observed that a unit Killing vector field ξ on an n-dimensional Riemannian manifold (M,g), influences its algebra of smooth functions C∞(M) For instance, if h is an eigenfunction of the Laplace operator Δ with eigenvalue λ, then ξ(h) is also eigenfunction with same eigenvalue Additionally, it has been observed that the Hessian Hh(ξ,ξ) of a smooth function h∈C∞(M) defines a self adjoint operator ⊡ξ and has properties similar to most of properties of the Laplace operator on a compact Riemannian manifold (M,g) We study several properties of functions associated to the unit Killing vector field ξ Finally, we find characterizations of the odd dimensional sphere using properties of the operator ⊡ξ and the nontrivial solution of Fischer–Marsden differential equation, respectively

https://www.mdpi.com/2227-7390/9/4/307/pdf

A Note on Killing Calculus on Riemannian Manifolds

We use the fact that the tangent bundle TM of an orientable hypersurface M in the Euclidean space R n+1 is a submanifold of the Euclidean space R 2n+2, and use the induced metric on TM as submanifold to study its geometry. This induced metric is not a natural metric in the sense that the projection π : TM → M is not a Riemannian submersion (which holds for Sasaki and other metrics, used to study geometry of the tangent bundle). First we prove that there is a reduction in the codimension of the submanifold TM and thus the tangent bundle TM is a hypersurface of the Euclidean space R 2n+1. As a consequence of our study, we infer that the induced metric on TS n the tangent bundle of the unit sphere S n makes TS n a Riemannian manifold of nonnegative sectional curvature. We also obtain a condition under which the tangent bundle TM of a hypersurface M in a Euclidean space is trivial.

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Tangent bundle of a hypersurface of a Euclidean space

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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Suha B. Al-Shaikh

Papers

Some Conditions on Trans-Sasakian Manifolds to Be Homothetic to Sasakian Manifolds

Tangent Bundle of the Hypersurfaces in a Euclidean Space

A Note on Killing Calculus on Riemannian Manifolds

Tangent bundle of a hypersurface of a Euclidean space

A Numerical Approach of Handling Fractional Stochastic Differential Equations