Beijing Institute of Mathematical Sciences and Applications

12th June ~ 12th July, 2024

Graduate level knowledge in differential geometry and in Riemannian geometry, and basic knowledge in partial differential equations and analysis.

1. Reminders of pre-requisites:
- Elements of differential geometry: manifold, tensor, metric, Lorentzian manifold, covariant derivative, Lie derivative, connection of Levi-Civita, Christoffel symbols, Riemann tensor, Ricci tensor, hypersurface, second fundamental form, Minkowski metric.
2. The Einstein-Maxwell and the Einstein-Yang-Mills equations:
- General Relativity.
- The Yang-Mills equations: the Yang-Mills curvature, the Bianchi identities, the Yang-Mills stress-energy-momentum tensor, the Yang-Mills field equations.
- The Einstein-Yang-Mills system.
3. The Cauchy problem and the constraints for the Einstein-Yang-Mills system:
- The Cauchy problem.
- The constraint equations.
4. The gauges invariance of the equations:
- The invariance under gauge transformation.
- The diffeomorphism invariance.
5. The Einstein-Yang-Mills equations as a system on the perturbation of Minkowski space-time:
- Looking at the metric as a perturbation of the Minkowski space-time.
- The Einstein-Yang-Mills equations in a given system of coordinates.
6. The Einstein-Yang-Mills system in the Lorenz gauge and in wave coordinates as a non-linear hyperbolic system of partial differential equations.
7. Construction of the initial data and the gauges conditions constraints:
- The initial data for the Yang-Mills potential.
- The initial data for the metric.
- The propagation of the Lorenz gauge condition.
- The propagation of the wave coordinates condition.
- Construction of the initial data for the hyperbolic system given an initial data set that solves the Einstein-Yang-Mills constraints.

[1] L. Andersson and V. Moncrief, Future complete vacuum spacetimes, in The Einstein equations and the large scale behavior of gravitational fields, Birkhäuser, Basel (2004).
[2] Y. C. Bruhat, Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non-linéaires., Acta Math. 88, 1952, 141–225.
[3] Y. C. Bruhat, R. Geroch, Global Aspects of the Cauchy Problem in General Relativity. Comm. Math. Phys. 14, 1969, 329–335.
[4] M. P. Do Carmo, Riemannian Geometry, Birkhäuser (1992).
[5] D. Eardley, V. Moncrief, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness properties, Comm. Math. Phys. 83 (1982), no. 2, 171-191.
[6] D. Eardley, V. Moncrief, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. II. Completion of proof, Comm. Math. Phys. 83 (1982), no. 2, 193-212.
[7] A. Einstein, Zur Elektrodynamik bewegter Körper, Annalen der Physik und Chemie 17, 1905, 891– 921.
[8] A. Einstein, Der Feldgleichungen des Gravitation, Preuss. Akad. Wiss. Berlin, Sitzber., 1915, 844– 847.
[9] S. W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space-Time Cambridge University Press, 1973.
[10] J. Isenberg, The Initial Value Problem in General Relativity, in: Ashtekar, A., Petkov, V. (eds) Springer Handbook of Spacetime. Springer Handbooks. Springer, Berlin, Heidelberg (2014).
[11] H. Minkowski, “Raum und Zeit”, Physikalische Zeitschrift, 10. Jahrgang, 1909, 104–115.
[12] P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171, Springer, 1998.
[13] B. Riemann, Über die Hypothesen, welche der Geometrie zugrunde liegen, Habilitationsschrift, 1854, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13 (1868).
[14] H. Ringström, The Cauchy Problem in General Relativity, ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich, 2009.
[15] C. N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev., 96:191–195, Oct 1954.
[16] R. Wald, General Relativity The University of Chicago Press, 1984.

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