Abstract
In this paper, we introduce a geometric structure that is capable of describing matter and forces simultaneously. This structure can be established by using the notion of $$Z_{2}$$ -graded Lie algebroid structures and graded semi-Riemannian metrics on them. Using calculus of variations, we derive field equations from the extended Hilbert–Einstein action. The derived equations contain Yang–Mills and Einstein field equations simultaneously. The even part of the graded Lie algebroid describes forces and its odd part is related to matter and particles.
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