On Some Geometric Structures Associated to a k-Symplectic Manifold

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A canonical connection is attached to any k-symplectic manifold. We study the properties of this connection and its geometric applications to k-symplectic manifolds. In particular we prove that, under some natural assumption, any ksymplectic manifold admits an Ehresmann connection, discussing some corollaries of this result, and we find vanishing theorems for characteristic classes on a k-symplectic manifold.