[Paper Review] Yamabe and quasi-Yamabe solitons on Euclidean submanifolds

Bang‐Yen Chen, Sharief Deshmukh|arXiv (Cornell University)|Nov 8, 2017

Geometric Analysis and Curvature Flows5 references3 citations

TL;DR

This paper investigates Yamabe and quasi-Yamabe solitons on Euclidean submanifolds where the soliton vector field is the tangential component of the position vector field. It establishes that such solitons on proper hypersurfaces in Euclidean space are quasi-umbilical with the tangential position field as the distinguished direction, and when μ ≠ 0, the hypersurface must be an open portion of a rotational hypersurface whose axis passes through the origin.

ABSTRACT

In this paper we initiate the study of Yamabe and quasi-Yamabe solitons on Euclidean submanifolds whose soliton fields are the tangential components of their position vector fields. Several fundamental results of such solitons were proved. In particular, we classify such Yamabe and quasi-Yamabe solitons on Euclidean hypersurfaces.

Motivation & Objective

• To initiate the study of Yamabe and quasi-Yamabe solitons on Euclidean submanifolds where the soliton field is the tangential component of the position vector field.
• To classify such solitons on Euclidean hypersurfaces, particularly under geometric constraints involving the position vector field.
• To explore the geometric implications of the soliton condition when the soliton field is derived from the position vector field.
• To establish a link between the soliton structure and intrinsic geometric properties such as quasi-umbilicity and torse-forming vector fields.
• To determine the necessary and sufficient conditions under which a hypersurface in Euclidean space admits a quasi-Yamabe soliton with the tangential position field as the soliton vector field.

Proposed method

• Utilizes the Lie derivative of the metric to define Yamabe and quasi-Yamabe solitons via the equation $\frac{1}{2}\mathcal{L}_X g = (R - \lambda)g$ and $\frac{1}{2}\mathcal{L}_X g = (R - \lambda)g + \mu X^\# \otimes X^\#$, respectively.
• Applies the Gauss and Weingarten formulas to relate the Levi-Civita connection on the submanifold to the ambient Euclidean connection.
• Derives the key condition $g(A_{{\bf x}^N}V, W) = (R - \lambda - 1)g(V,W) + \mu g({\bf x}^T, V)g({\bf x}^T, W)$ from the soliton equation and the geometry of the position vector field.
• Employs the concept of torse-forming vector fields, defined by $\nabla_X v = \varphi X + \alpha(X)v$, to analyze the geometric behavior of $\mathbf{x}^T$.
• Uses the classification of rotational hypersurfaces and the structure of cones and spheres to characterize the possible soliton solutions.
• Applies the condition that $\mathbf{x}^T$ is nowhere zero (proper submanifold) to restrict the class of admissible hypersurfaces.

Experimental results

Research questions

• RQ1 Under what conditions does a Euclidean submanifold admit a Yamabe soliton with the tangential component of the position vector field as the soliton field?
• RQ2 What geometric structure must a hypersurface in $\mathbb{E}^{n+1}$ possess to support a quasi-Yamabe soliton with $\mu \neq 0$ and soliton field $\mathbf{x}^T$?
• RQ3 How does the tangential component of the position vector field relate to the shape operator and scalar curvature in the context of soliton equations?
• RQ4 Can the soliton field $\mathbf{x}^T$ be a torse-forming vector field, and what does this imply about the hypersurface's geometry?
• RQ5 What class of hypersurfaces in $\mathbb{E}^{n+1}$ supports a quasi-Yamabe soliton with $\mathbf{x}^T$ as the soliton field when $\mu \neq 0$?

Key findings

• A Euclidean hypersurface $M$ in $\mathbb{E}^{n+1}$ admits a quasi-Yamabe soliton with soliton field $\mathbf{x}^T$ and $\mu \neq 0$ if and only if $M$ is a quasi-umbilical hypersurface with $\mathbf{x}^T$ as its distinguished direction.
• The tangential component $\mathbf{x}^T$ of the position vector field is a torse-forming vector field on $M$ under the quasi-Yamabe soliton condition with $\mu \neq 0$.
• If $M$ is a proper hypersurface and $\mu \neq 0$, then $M$ must be an open portion of a rotational hypersurface whose axis of rotation passes through the origin.
• The soliton condition leads to a specific form of the shape operator $A_{{\bf x}^N}$, which acts as a scalar multiple of the identity on the orthogonal complement of $\mathbf{x}^T$ and has a distinct action along $\mathbf{x}^T$.
• The scalar curvature $R$ and the soliton constant $\lambda$ are related to the geometry of the hypersurface through the equation $g(A_{{\bf x}^N}V, W) = (R - \lambda - 1)g(V,W) + \mu g({\bf x}^T, V)g({\bf x}^T, W)$.
• The classification result implies that no hypersurface that is an open portion of a hypersphere, spherical cylinder, or spherical cone can support such a quasi-Yamabe soliton with $\mu \neq 0$ unless it is excluded by the hypothesis.