[Paper Review] Harmonic Maps between Generalized Lagrange Spaces

Mircea Neagu|arXiv (Cornell University)|Sep 14, 2000

Advanced Differential Geometry Research6 references3 citations

TL;DR

This paper introduces a generalized framework for harmonic maps between generalized Lagrange spaces using a connection tensor $ P $, extending classical harmonic map theory to include gravitational and electromagnetic structures. It proves that solutions to certain systems of PDEs—particularly those modeling electrodynamics and gravity—correspond to harmonic maps in this geometry, unifying geometric and physical field theories via energy functionals derived from Lagrangian metrics.

ABSTRACT

In this paper are studied the harmonic maps between two generalized Lagrange spaces. At the same time, it is proved that the solutions of $C^2$ class of certain ODEs or PDEs are harmonic maps between certain convenient generalized Lagrange spaces.

Motivation & Objective

• To define harmonic maps between generalized Lagrange spaces using a connection tensor $ P $ and a $ (g, \varphi, h) $-energy functional.
• To extend classical harmonic map theory to include non-Riemannian structures with gravitational and electromagnetic potentials.
• To show that solutions of specific systems of PDEs (e.g., in electrodynamics and relativistic field theory) are critical points of the energy functional, i.e., harmonic maps.
• To construct a geometric framework where $ C^2 $ solutions of PDEs naturally arise as harmonic maps in a generalized Lagrange space.
• To explore the possibility of a unique geometric structure associated with a given PDE system, posing an open problem for future work.

Proposed method

• Define generalized Lagrange spaces via fundamental metrics $ g_{\alpha\beta}(a,b) $ and $ h_{ij}(x,y) $, where $ b $ and $ y $ are fiber coordinates on tangent bundles.
• Introduce a connection tensor $ P^{\beta}_{\alpha i} $ and $ P^{j}_{\alpha i} $ to relate the metric structures of $ M $ and $ N $, enabling the construction of the energy functional.
• Construct the $ (g, \varphi, h)^P $-energy functional: $ E^P_{g\varphi h}(f) = \frac{1}{2}\int_M g^{\alpha\beta} h_{ij} f^i_\alpha f^j_\beta \sqrt{\varphi} da $, with $ b $ and $ y $ defined via $ P $ and the Riemannian metric $ \varphi_{\alpha\beta} $.
• Derive the Euler-Lagrange equations for the energy functional, yielding the harmonic map equations involving Christoffel symbols and curvature terms.
• Apply the framework to Lagrange spaces of electrodynamics, where $ L_M $ and $ L_N $ include gravitational, electromagnetic, and potential terms, leading to a simplified energy functional independent of $ P $.
• Show that in relativistic models with $ g_{ij}(x,y) = e^{2\sigma(x)}\varphi_{ij}(x) $, the Einstein equations are modified by an additional tensor $ t_{ij} $, and Maxwell’s equations are recovered under specific conditions.

Experimental results

Research questions

• RQ1 Can harmonic maps be defined between generalized Lagrange spaces using a connection tensor $ P $, and how does this generalize classical harmonic maps?
• RQ2 Do solutions of systems of PDEs in electrodynamics and general relativity correspond to harmonic maps in a generalized Lagrange geometry?
• RQ3 How does the energy functional depend on the connection tensor $ P $, and when is it independent of $ P $, as in the case of electrodynamics?
• RQ4 What is the geometric and physical interpretation of the additional tensor $ t_{ij} $ in the Einstein equations derived from a generalized Lagrange metric?
• RQ5 Is it possible to associate a unique generalized Lagrange geometry to a given system of PDEs, or is the construction inherently non-unique?

Key findings

• Solutions to systems of PDEs, including those in electrodynamics and relativistic field theory, are shown to be critical points of the $ (g, \varphi, h)^P $-energy functional, i.e., harmonic maps.
• In the case of Lagrange spaces of electrodynamics, the energy functional simplifies to $ E = \frac{1}{2}\int_M g^{\alpha\beta} h_{ij} f^i_\alpha f^j_\beta \sqrt{\varphi} da $, independent of the connection tensor $ P $, and the Euler-Lagrange equations yield harmonic map equations with curvature and metric terms.
• For a relativistic model with $ g_{ij}(x,y) = e^{2\sigma(x)}\varphi_{ij}(x) $, the Einstein equations are modified by a tensor $ t_{ij} $, which includes terms involving $ \sigma_{ij} $, curvature, and derivatives of $ \sigma $, showing a non-classical correction to gravity.
• When $ \sigma = \sigma(x) $, the $ v $-electromagnetic tensor vanishes, the $ h $-covariant derivative reduces to the Levi-Civita connection, and Maxwell’s equations reduce to the classical form.
• The framework unifies harmonic maps, PDE solutions, and field theories in a single geometric setting, suggesting that $ C^2 $ solutions of PDEs naturally arise as harmonic maps in generalized Lagrange spaces.
• The paper poses an open problem: whether a unique generalized Lagrange geometry can be canonically associated with a given PDE system, with a future paper promising a solution via higher-order jet fibrations.