In this paper, we will give the definition of the pedal curves of frontals and investigate the geometric properties of these curves in the Euclidean plane. We obtain that pedal curves of frontals in the Euclidean plane are also frontals. We further discuss the connections between singular points of the pedal curves and inflexion points of frontals in the Euclidean plane.
Ricci solitons (RS) have an extensive background in modern physics and are extensively used in cosmology and general relativity. The focus of this work is to investigate Ricci almost solitons (RAS) on Lorentzian manifolds with a special metric connection called a semi-symmetric metric u-connection (SSM-connection). First, we show that any quasi-Einstein Lorentzian manifold having a SSM-connection, whose metric is RS, is Einstein manifold. A similar conclusion also holds for a Lorentzian manifold with SSM-connection admitting RS whose soliton vector Z is parallel to the vector u. Finally, we examine the gradient Ricci almost soliton (GRAS) on Lorentzian manifold admitting SSM-connection.
Notions of the pedal curves of regular curves are classical topics. T. Nishimura [T. Nishimura, Demonstratio Math., 43 (2010), 447-459] has done some work associated with the singularities of pedal curves of regular curves. But if the curve has singular points, we can not define the Frenet frame at these singular points. We also can not use the Frenet frame to define and study the pedal curve of the original curve. In this paper, we consider the differential geometry of pedal curves of singular curves in the sphere. We define the pedal curve of a front and give properties of such pedal curve by using a moving frame along a front. At last, we give the classification of singularities of the pedal curves of fronts.
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