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Curvature tensors and Ricci solitons with respect to Zamkovoy connection in anti-invariant submanifolds of trans-Sasakian manifold

Payel Karmakar (2022)

Mathematica Bohemica

The present paper deals with the study of some properties of anti-invariant submanifolds of trans-Sasakian manifold with respect to a new non-metric affine connection called Zamkovoy connection. The nature of Ricci flat, concircularly flat, ξ -projectively flat, M -projectively flat, ξ - M -projectively flat, pseudo projectively flat and ξ -pseudo projectively flat anti-invariant submanifolds of trans-Sasakian manifold admitting Zamkovoy connection are discussed. Moreover, Ricci solitons on Ricci flat,...

Curvatures of the diagonal lift from an affine manifold to the linear frame bundle

Oldřich Kowalski, Masami Sekizawa (2012)

Open Mathematics

We investigate the curvature of the so-called diagonal lift from an affine manifold to the linear frame bundle LM. This is an affine analogue (but not a direct generalization) of the Sasaki-Mok metric on LM investigated by L.A. Cordero and M. de León in 1986. The Sasaki-Mok metric is constructed over a Riemannian manifold as base manifold. We receive analogous and, surprisingly, even stronger results in our affine setting.

Deformations of Metrics and Biharmonic Maps

Aicha Benkartab, Ahmed Mohammed Cherif (2020)

Communications in Mathematics

We construct biharmonic non-harmonic maps between Riemannian manifolds ( M , g ) and ( N , h ) by first making the ansatz that ϕ : ( M , g ) → ( N , h ) be a harmonic map and then deforming the metric on N by h ˜ α = α h + ( 1 - α ) d f ⊗ d f to render ϕ biharmonic, where f is a smooth function with gradient of constant norm on ( N , h ) and α ∈ ( 0 , 1 ) . We construct new examples of biharmonic non-harmonic maps, and we characterize the biharmonicity of some curves on Riemannian manifolds.

Diastolic and isoperimetric inequalities on surfaces

Florent Balacheff, Stéphane Sabourau (2010)

Annales scientifiques de l'École Normale Supérieure

We prove a universal inequality between the diastole, defined using a minimax process on the one-cycle space, and the area of closed Riemannian surfaces. Roughly speaking, we show that any closed Riemannian surface can be swept out by a family of multi-loops whose lengths are bounded in terms of the area of the surface. This diastolic inequality, which relies on an upper bound on Cheeger’s constant, yields an effective process to find short closed geodesics on the two-sphere, for instance. We deduce...

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