Integral Identities and Rigidity of Generalized $m$-Quasi-Einstein Manifolds

We establish differential and integral identities for closed generalized $m$-quasi-Einstein manifolds $(M^n,g,X,\lambda )$. These identities yield criteria for conformality, the Killing condition, and triviality, and provide a unified framework for several rigidity phenomena. Afther this, we recast the integrated Bochner formula as a Witten-type Hodge-energy identity. The cancellation of its quartic term at $m=-2$ leads to a triviality theorem in the generalized setting under a natural sign condition and, in particular, proves a recently Colling–Dunajski conjecture at $m=-2$ when $\lambda \le 0$ is constant. An identity for the drift laplacian gives a new proof of the previously known triviality result for $m\le -n$ and extends it to generalized $m$-quasi-Einstein manifolds. In the constant-$\lambda$ case, we then derive a differential identities for twisted operators that singles out the value $m=-4$. Combined with spectral and maximum-principle arguments, this yields triviality for $\lambda \le 0$, thereby proving the Colling–Dunajski conjecture at $m=-4$. Finally, complementing our criterion characterizing when a conformal potential field is Killing, we exhibit in the appendix a closed generalized $m$-quasi-Einstein manifold whose potential field is conformal but non-Killing.