SUMMARY
The discussion focuses on the characterization of diffeomorphisms on a Riemannian manifold (M,g) that preserve a Killing vector field, denoted as ##\xi##, where the Lie derivative of the metric tensor remains zero, ##\mathcal{L}_{\xi} g = 0##. The group of diffeomorphisms ##f: M \to M## that satisfy the condition ##\mathcal{L}_{f^* \xi} (f^*g) = 0## are identified as symmetry-preserving transformations. These transformations maintain the Killing vector property under the pullback of the metric, ensuring that ##\xi## remains a Killing vector for the transformed metric ##f^* g##.
PREREQUISITES
- Understanding of Riemannian geometry and manifolds
- Familiarity with Killing vector fields and their properties
- Knowledge of Lie derivatives and their applications in differential geometry
- Basic concepts of diffeomorphisms and their role in geometric transformations
NEXT STEPS
- Research the properties of Killing vector fields in Riemannian geometry
- Study the implications of Lie derivatives on geometric structures
- Explore the classification of symmetry groups in differential geometry
- Investigate examples of diffeomorphisms that preserve Killing vectors in specific manifolds
USEFUL FOR
Mathematicians, theoretical physicists, and researchers in differential geometry focusing on symmetries and geometric transformations in Riemannian manifolds.