The author considers the Nijenhuis map assigning to two type (1,1) tensor fields $\alpha$, $\beta$ a mapping $$\langle \alpha ,\beta \rangle :(\xi ,\zeta )\mapsto \left[\alpha \right(\xi ),\beta (\zeta \left)\right]+\alpha \circ \beta \left(\right[\xi ,\zeta \left]\right)-\alpha \left(\right[\xi ,\beta \left(\zeta \right)\left]\right)-\beta \left(\right[\alpha \left(\xi \right),\zeta \left)\right]),$$ where $\xi$, $\zeta$ are vector fields. Then $\langle \alpha ,\beta \rangle$ is a type (2,1) tensor field (Nijenhuis tensor) if and only if $[\alpha ,\beta ]=0$. Considering a smooth manifold $X$ with a smooth action of a Lie group, a secondary invariant may be defined as a mapping whose area of invariance is restricted to the inverse image of an invariant subset of $X$ under another invariant mapping. The author recognizes a secondary invariant related to the...

[For the entire collection see Zbl 0699.00032.] A manifold (M,g) is said to be generalized Einstein manifold if the following condition is satisfied $$\left({\nabla }_{X}S\right)(Y,Z)=\sigma \left(X\right)g(Y,Z)+\nu \left(Y\right)g(X,Z)+\nu \left(Z\right)g(X,Y)$$ where S(X,Y) is the Ricci tensor of (M,g) and $\sigma$ (X), $\nu$ (X) are certain $\ell$-forms. In the present paper the author studies properties of conformal and geodesic mappings of generalized Einstein manifolds. He gives the local classification of generalized Einstein manifolds when g($\psi$ (X),$\psi$ (X))$\ne 0$.