[Paper Review] Full description of totally geodesic unit vector fields on 2-dimensional Riemannian manifolds

Alexander Yampolsky|ArXiv.org|Mar 24, 2005

Geometric Analysis and Curvature Flows19 references3 citations

TL;DR

This paper provides a complete geometric classification of totally geodesic unit vector fields on 2-dimensional Riemannian manifolds by analyzing their image as submanifolds in the unit tangent bundle equipped with the Sasaki metric. It shows that such fields exist if and only if the metric is locally of the form $ ds^2 = du^2 + \sin^2\alpha(u)\,dv^2 $, where $ \alpha(u) $ satisfies a specific ODE, and the vector field has a prescribed angular dependence on $ v $, with explicit solutions for all real parameters $ a $ and $ \omega_0 $. The key contribution is a full characterization of the metric and vector field structure for which the image is totally geodesic.

ABSTRACT

We give a full geometrical description of local totally geodesic unit vector field on Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its unit tangent bundle with the Sasaki metric.

Motivation & Objective

• To provide a complete geometric description of locally totally geodesic unit vector fields on 2-dimensional Riemannian manifolds.
• To determine the necessary and sufficient conditions on the metric and vector field for the image of the field in the unit tangent bundle to be totally geodesic under the Sasaki metric.
• To characterize all such vector fields explicitly in terms of a differential equation for the function $ \alpha(u) $, and to describe their geometric behavior in terms of curvature and angular velocity.
• To establish conditions under which the metric admits an isometric immersion into Euclidean 3-space as a surface of revolution.

Proposed method

• The paper models the unit vector field $ \xi $ as a mapping $ \xi: M^2 \to T_1M^2 $, treating its image as a submanifold in the unit tangent bundle equipped with the Sasaki metric.
• It uses an orthonormal frame $ \{e_0, e_1\} $ adapted to the vector field and its orthogonal complement, introducing angle functions and curvature components to express the covariant derivative structure.
• The condition for total geodesy is reduced to a system of PDEs involving the geodesic curvatures $ k $ and $ \varkappa $, which are related to the angle function $ \omega $ and the metric function $ \alpha(u) $.
• By assuming a metric of the form $ ds^2 = du^2 + \sin^2\alpha(u)\,dv^2 $, the problem reduces to solving a first-order ODE: $ \frac{d\alpha}{du} = 1 - \frac{a+1}{\cos\alpha} $, where $ a $ is a constant parameter.
• The vector field is constructed explicitly as $ \xi = \cos(av + \omega_0)\partial_u + \frac{\sin(av + \omega_0)}{\sin\alpha(u)}\partial_v $, which is parallel along meridians and rotates with constant angular speed $ a $ along parallels.
• The paper derives intrinsic equations for the integral curves of $ \xi $, relating the geodesic curvature $ k $ and normal curvature $ \varkappa $ through a first integral, and analyzes the existence of isometric immersions into $ \mathbb{E}^3 $ via surfaces of revolution.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions on a 2-dimensional Riemannian metric for the existence of a locally totally geodesic unit vector field in the Sasaki metric on the unit tangent bundle?
• RQ2 How can such a vector field be explicitly constructed in terms of geometric and differential invariants of the base manifold?
• RQ3 What is the geometric behavior of the integral curves of a totally geodesic unit vector field, and how does it relate to curvature and symmetry?
• RQ4 Under what conditions does the metric admit an isometric immersion into Euclidean 3-space as a surface of revolution?
• RQ5 What role does the parameter $ a $ play in determining the topology and curvature of the integral curves of the vector field?

Key findings

• A unit vector field $ \xi $ is totally geodesic in the Sasaki metric on $ T_1M^2 $ if and only if the metric is locally of the form $ ds^2 = du^2 + \sin^2\alpha(u)\,dv^2 $, where $ \alpha(u) $ satisfies $ \frac{d\alpha}{du} = 1 - \frac{a+1}{\cos\alpha} $ for some constant $ a $.
• The totally geodesic unit vector field is explicitly given by $ \xi = \cos(av + \omega_0)\partial_u + \frac{\sin(av + \omega_0)}{\sin\alpha(u)}\partial_v $, with $ a, \omega_0 \in \mathbb{R} $ constants.
• When $ a = -1 $, the integral curves of $ \xi $ are circles on the unit sphere, and their preimages under stereographic projection are straight lines in the plane, with the sphere realized as a surface of revolution.
• For $ a = -1 $, the metric corresponds to the standard metric on the 2-sphere, and the vector field corresponds to a family of circles passing through the south pole.
• The metric admits an isometric immersion into $ \mathbb{E}^3 $ as a surface of revolution only if $ |a+1| < 1 $, i.e., $ -2 < a < 0 $, with the immersion defined via $ x(\alpha) = \sin\alpha $, $ z(\alpha) = \int \frac{\cos t}{a+1 - \cos t} \sqrt{1 - (a+1 - \cos t)^2} \, dt $.
• The intrinsic equation of the integral curves is expressed as $ k = c\big{[}a + (a+2)(k^2 + \varkappa^2)\big{]}^{\frac{a+1}{a+2}} $ for $ a \neq 0, -2 $, showing a precise algebraic relation between curvature and angular velocity.