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Conformal geodesics are not variational in higher dimensions

Boris Kruglikov

TL;DR

The paper addresses whether conformal geodesic equations admit a conformally invariant variational formulation in dimensions $n>3$. Using the inverse problem of the calculus of variations, jet-space analysis, and dimension-reduction techniques, it establishes that both parametrized and unparametrized conformal geodesics are not variational in higher dimensions, with the 3D case remaining special (unparametrized conformal circles are variational via a torsion-based Lagrangian). The argument combines explicit flat-model obstructions (via Vainberg–Tonti reduction) with a generic-rank analysis to extend non-variationality to general metrics. This leads to a proposed selection principle: variationality of conformal geodesics singles out dimension three, with important implications for the role of action principles in conformal geometry and general relativity.

Abstract

Variationality of the equation of conformal geodesics is an important problem in geometry with applications to general relativity. Recently it was proven that, in three dimensions, this system of equations for un-parametrized curves is the Euler-Lagrange equations of a certain conformally invariant functional, while the parametrized system in three dimensions is not variational. We demonstrate that variationality fails in higher dimensions for both parametrized and un-parametrized conformal geodesics, indicating that variational principle may be the selection principle for the physical dimension.

Conformal geodesics are not variational in higher dimensions

TL;DR

The paper addresses whether conformal geodesic equations admit a conformally invariant variational formulation in dimensions . Using the inverse problem of the calculus of variations, jet-space analysis, and dimension-reduction techniques, it establishes that both parametrized and unparametrized conformal geodesics are not variational in higher dimensions, with the 3D case remaining special (unparametrized conformal circles are variational via a torsion-based Lagrangian). The argument combines explicit flat-model obstructions (via Vainberg–Tonti reduction) with a generic-rank analysis to extend non-variationality to general metrics. This leads to a proposed selection principle: variationality of conformal geodesics singles out dimension three, with important implications for the role of action principles in conformal geometry and general relativity.

Abstract

Variationality of the equation of conformal geodesics is an important problem in geometry with applications to general relativity. Recently it was proven that, in three dimensions, this system of equations for un-parametrized curves is the Euler-Lagrange equations of a certain conformally invariant functional, while the parametrized system in three dimensions is not variational. We demonstrate that variationality fails in higher dimensions for both parametrized and un-parametrized conformal geodesics, indicating that variational principle may be the selection principle for the physical dimension.
Paper Structure (5 sections, 6 theorems, 37 equations)

This paper contains 5 sections, 6 theorems, 37 equations.

Table of Contents

  1. Formulation of the Problem and the Result
  2. Warm-up: variational calculus and circles in three and two dimensions
  3. The inverse problem for conformal circles in four and five dimensions
  4. Reduction and Proof in the general case
  5. Discussion: on the selection principle

Key Result

Theorem 1

Let $(M^n,[g])$ be a conformal manifold. If its dimension $n>3$, then the equation for conformal geodesics is not variational in the classical sense neither in parametrized nor in unparametrized form.

Theorems & Definitions (12)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Remark 3
  • Proposition 4
  • ...and 2 more