[Paper Review] Metallic maps between metallic Riemannian manifolds and constancy of certain maps

Mehmet Akif Akyol|arXiv (Cornell University)|Feb 10, 2017

Geometric Analysis and Curvature Flows12 references3 citations

TL;DR

This paper introduces metallic maps between metallic Riemannian manifolds by imposing a holomorphic-like condition, establishing sufficient conditions for such maps to be harmonic or totally geodesic. The key contribution is proving that under specific compatibility conditions with almost complex, almost contact, or para-contact structures, these maps are necessarily constant, with a critical exception in the copper case (p=1, q=2).

ABSTRACT

In this paper, we introduce metallic maps between metallic Riemannian manifolds, provide an example and obtain certain conditions for such maps to be totally geodesic. We also give a sufficient condition for a map between metallic Riemannian manifolds to be harmonic map. Then we investigate the constancy of certain maps between metallic Riemannian manifolds and various manifolds by imposing the holomorphic-like condition. Moreover, we check the reverse case and show that some such maps are constant if there is a condition for this.

Motivation & Objective

• To define and study metallic maps between metallic Riemannian manifolds using a holomorphic-like condition.
• To establish sufficient conditions for such maps to be harmonic or totally geodesic.
• To investigate when metallic maps to or from manifolds with almost complex, almost contact, or para-contact structures are constant.
• To determine the conditions under which such maps are forced to be constant, including identifying exceptional cases.

Proposed method

• Introduce metallic maps between metallic Riemannian manifolds by requiring the differential of the map to commute with the metallic structure tensor field $J$.
• Use the defining equation $J^2 = pJ + qI$ for metallic structures, where $p, q$ are positive integers.
• Apply the holomorphic-like condition $F_* J_1 = J_2 F_*$ between manifolds $(M_1, g_1, J_1)$ and $(M_2, g_2, J_2)$ to derive compatibility relations.
• Derive linear dependence relations between $F_*(X)$, $F_*(J_1X)$, and the Reeb vector field $\xi$ using the structure equations of the target manifolds.
• Use repeated application of the structure endomorphisms ($J$, $\phi$, $\phi'$) and the metric compatibility conditions to derive vector field identities.
• Show that the resulting system of equations forces $F_*(X) = 0$ for all $X$, implying $F$ is constant, except when $p^2 = (q-1)^2$ in the para-contact case.

Experimental results

Research questions

• RQ1 Under what conditions is a metallic map between metallic Riemannian manifolds harmonic?
• RQ2 When is a metallic map between metallic Riemannian manifolds totally geodesic?
• RQ3 When does a smooth map from a metallic Riemannian manifold to an almost contact metric manifold satisfy $F_* J_1 = \phi F_*$ and remain non-constant?
• RQ4 What is the role of the parameters $p$ and $q$ in determining the constancy of such maps?
• RQ5 Are there exceptions to the constancy result when the target manifold is para-contact metric?

Key findings

• A smooth map $F$ from a metallic Riemannian manifold $(M_1, g_1, J_1)$ to an almost complex manifold $(M', J')$ satisfying $F_* J_1 = J' F_*$ is constant.
• A smooth map $F$ from a metallic Riemannian manifold $(M_1, g_1, J_1)$ to an almost contact metric manifold $(M, \phi, \xi, \eta, g)$ satisfying $F_* J_1 = \phi F_*$ is constant.
• A smooth map $F$ from a metallic Riemannian manifold $(M_1, g_1, J_1)$ to an almost para-contact metric manifold $(N', \phi', \xi', \eta', g')$ satisfying $F_* J_1 = \phi' F_*$ is constant if $p^2 \neq (q-1)^2$.
• The exception to the constancy result occurs when $p^2 = (q-1)^2$, which includes the copper case ($p=1, q=2$), allowing for the possibility of non-constant maps.
• The constancy of such maps is derived from a system of vector field identities that force $F_*(X) = 0$ after repeated application of the structure endomorphisms and metric compatibility.
• The paper establishes that the holomorphic-like condition $F_* J_1 = J_2 F_*$ on metallic structures leads to linear dependence that collapses the differential to zero, implying the map is constant.