Paper · Aug 16, 2025

Almost Ricci Solitons on Class $A$ Hypersurfaces of Product Spaces

Ahmet Umut Çoraplı, Burcu Bektaş Demirci, Nurettin Cenk Turgay

math.DGarXiv:2508.12098

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Abstract

In this paper, we study hypersurfaces in the product spaces $Q_{ε}^{3} \times R$ for which the tangential component $T$ of the vector field $\frac{\partial}{\partial t}$ is a principal direction, where $Q_{ε}^{3}$ denotes the three-dimensional non-flat Riemannian space form with sectional curvature $ε = \pm 1$ , and $\frac{\partial}{\partial t}$ is the unit vector field tangent to the $R$ -factor. We obtain a local classification of hypersurfaces with three distinct principal curvatures satisfying specific functional relations. Then, we determine the necessary and sufficient conditions for such hypersurfaces to admit an almost Ricci soliton structure with potential vector field $T$ . Finally, we prove that the only hypersurfaces admitting such solitons are rotational, by showing that the constructed examples with three distinct principal curvatures do not admit almost Ricci solitons.

Ricci solitons Almost & η-Ricci solitons 3-manifolds Sphere theorems

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Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Almost & η-Ricci solitons Variations on the Ricci soliton equation: let the constant λ become a function (almost solitons), add a term along a contact form (η-Ricci… Start with: Ricci almost solitons (Pigola et al., 2010) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

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11A.11
Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles
Positive Ricci curvature in dimension 3 flows to a round metric.
10A.1
A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization
The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.
10A.2
Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization
Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

Almost Ricci Solitons on Class $A$ Hypersurfaces of Product Spaces

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Almost Ricci Solitons on Class A Hypersurfaces of Product Spaces

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Almost Ricci Solitons on Class $A$ Hypersurfaces of Product Spaces