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In this paper, we introduce the notion of Koszul-Vinberg-Nijenhuis structureson a left-symmetric algebroid as analogues of Poisson-Nijenhuis structures on aLie algebroid, and show that a Koszul-Vinberg-Nijenhuis structure gives rise toa hierarchy of Koszul-Vinberg structures. We introduce the notions of ${\rmKV\Omega}$-structures, pseudo-Hessian-Nijenhuis structures and complementarysymmetric $2$-tensors for Koszul-Vinberg structures on left-symmetricalgebroids, which are analogues of ${\rm P\Omega}$-structures,symplectic-Nijenhuis structures and complementary $2$-forms for Poissonstructures. We also study the relationships between these various structures.

Koszul-Vinberg structures and compatible structures on left-symmetric algebroids