Proportional derivative plus feed forward (PD+FF) feedback control laws exist to locally asymptotically track a smooth reference trajectory having bounded velocity for a fully actuated simple mechanical system (SMS) on a Riemannian manifold. Almost-global asymptotically stable (AGAS) tracking for an SMS on a compact Lie group is also addressed in the literature by observing that the closed loop error dynamics is obtained by lifting the gradient vector field of a navigation function and stabilizing the error dynamics about the Lie group identity. Almost-global setpoint stabilization for an SMS on a Riemannian manifold is often tackled by introducing a gradient field of a Morse function as a feedback control force for the dissipative SMS. In this paper we address the almost-global tracking problem for an SMS on a compact Riemannian manifold embedded in Euclidean space by explicitly introducing the error dynamics. A control law, including both feedback and feedforward terms, is then synthesized for the purpose of almost-global stabilization of the error dynamics.

Integral formulas in riemannian geometry

Aspects of globally-stable tracking for certain classes of simple mechanical systems

In this paper, we have studied almost Ricci-Yamabe soliton within the framework of certain classes of almost Kenmotsu manifold. First, we prove that complete Kenmotsu manifold admitting gradient almost Ricci-Yamabe soliton is locally isometric to hyperbolic space $\mathbb{H}^{2n+1}$ when $\xi r=0$. Next, we prove that almost Ricci-Yamabe soliton on Kenmotsu manifold reduces to Ricci-Yamabe soliton under the certain condition on the soliton function. Moreover, gradient almost Ricci-Yamabe soliton on $(\kappa,\mu)'$-almost Kenmotsu manifold and generalized $(\kappa,\mu)$-almost Kenmotsu manifold with $h'\neq0$ are examined. Some examples have been constructed of almost Ricci-Yamabe soliton on different class of warped product spaces.

Almost Ricci-Yamabe soliton on Almost Kenmotsu Manifolds

In the present paper, we give some characterizations by considering $*$-Ricci soliton as a Kenmotsu metric. We prove that if a Kenmotsu manifold represents an almost $*$-Ricci soliton with the potential vector field $V$ is a Jacobi along the Reeb vector field, then it is a steady $*$-Ricci soliton. Next, we show that a Kenmotsu matric endowed an almost $*$-Ricci soliton is Einstein metric if it is $\eta$-Einstein or the potential vector field $V$ is collinear to the Reeb vector field or $V$ is an infinitesimal contact transformation.

Geometry of almost contact metrics as almost $*$-Ricci solitons.

The object of the present paper is to characterize Sasakian 3-manifolds admitting a gradient Ricci-Yamabe soliton. It is shown that a Sasakian 3-manifold $M$ with constant scalar curvature admitting a proper gradient Ricci-Yamabe soliton is Einstein and locally isometric to a unit sphere. Also, the potential vector field is an infinitesimal automorphism of the contact metric structure. In addition, if $M$ is complete, then it is compact.

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Sasakian 3-manifolds admitting a gradient Ricci-Yamabe soliton

In this paper, via a new Hardy type inequality, we establish some cohomology vanishing theorems for free boundary compact submanifolds $M^n$ with $n\geq2$ immersed in the Euclidean unit ball $\mathbb{B}^{n+k}$ under one of the pinching conditions $|\Phi|^2\leq C$, $|A|^2\leq \widetilde{C}$, or $|\Phi|\leq R(p,|H|)$, where $A$ $(\Phi)$ is the (traceless) second fundamental form, $H$ is the mean curvature, $C,\widetilde{C}$ are positive constants and $R(p,|H|)$ is a positive function. In particular, we remove the condition on the flatness of the normal bundle, solving the first question, and partially answer the second question on optimal pinching constants proposed by Cavalcante, Mendes and Vitorio.

Integral formulas in riemannian geometry

Citations

Aspects of globally-stable tracking for certain classes of simple mechanical systems

Almost Ricci-Yamabe soliton on Almost Kenmotsu Manifolds

Geometry of almost contact metrics as almost $*$-Ricci solitons.

Sasakian 3-manifolds admitting a gradient Ricci-Yamabe soliton

Cohomology vanishing theorems for free boundary submanifolds

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