Universal curvature identities

We study scalar and symmetric 2-form valued universal curvature identities. We use this to establish the Gauss-Bonnet theorem using heat equation methods, to give a new proof of a result of Kuzʼmina and Labbi concerning the Euler-Lagrange equations of the Gauss-Bonnet integral, and to give a new derivation of the Euh-Park-Sekigawa identity.