Intrinsic General Relativity and Clifford Algebra

This paper proposes to go beyond the Einstein General Relativity theory in a noncommutative geometric framework. As first step, we rewrite intrinsically (coordinate-free formulation). second and Bianchi identities, including torsion, within minimal algebraic hypotheses. Then, order extend for dealing with scalar fields on manifold $$\mathcal {M}$$ , are forced adapt definition of vector connections. It leads consideration one associative product p four derivations ( $$\nabla $$ : (vector, vector) vector, $$\partial scalar) scalar, $$\delta (scalar, {C}$$ scalar). At last, particular case where scalars Clifford numbers motivates future investigations towards common writing field equations Dirac equation.