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A model of an extended manifold for the Dirac spinor field is considered. TwoLagrangians related by CPTM (charge-parity-time-mass) symmetry are constructedfor a pair of the Dirac spinor fields with each spinor field defined in aseparate manifold. An interaction between the matter fields in the manifolds isintroduced through gravity. A fermionic effective action of the general systemis constructed and a tadpole one-loop spinor diagram and part of the one-loopvacuum diagrams with two external gravitational off-shell fields whichcontribute to the effective action are calculated. It is demonstrated thatamong different versions of the second spinor Lagrangian there is a special onefor which a cancellation of the mentioned diagrams in the total effectiveaction takes place. As a result, the diagrams do not contribute to thecosmological constant, as well there is a zero contribution of the zero pointenergies of the spinor fields to the action. The non-zero leading order valueof the cosmological constant for each manifold in the framework is proportionalto the matter density of each separated manifold or difference of thedensities, depending on the chosen model of interaction of gravitational fieldswith fermions. An appearance of the dark matter in the model is shortlydiscussed as well as further applications of the approach.

Spinor fields, CPTM symmetry and smallness of cosmological constant in framework of extended manifold