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Motivated by the future stability problem of solutions of theEinstein--Yang--Mills (EYM) system with arbitrary dimension, we aim to (1)construct a tensorial symmetric hyperbolic formulation for the$(n+1)$-dimensional EYM system in the temporal gauge; (2) establish the localwell-posedness for the Cauchy problem of EYM equations in the temporal gaugeusing this tensorial symmetric hyperbolic system. By introducing certainauxiliary variables, we extend essentially the $(n+1)$-dimensional Yang--Millssystem to a tensorial symmetric hyperbolic system. On the contrary, thissymmetric hyperbolic system with data satisfying some constraints (extendingthe Yang--Mills constraints) reduces to the Yang--Mills system. Consequently,an equivalence between the EYM and the tensorial symmetric hyperbolic systemwith a class of specific data set is concluded. Furthermore, a generalsymmetric hyperbolic system over tensor bundles is studied, with which, weconclude the local well-posedness of the EYM system. It turns out the idea ofsymmetric hyperbolic formulation of the Yang--Mills field is very useful inprompting a tensorial Fuchsian formalism and proving the future stability forthe EYM system with arbitrary dimension (i.e., this new symmetric hyperbolicformulation of EYM manifests well behaved lower order terms for long timeevolution), see our companion article [21] with Todd A. Oliynyk.

A new symmetric hyperbolic formulation and the local Cauchy problem for the Einstein--Yang--Mills system in the temporal gauge