Bigness of tangent bundles and dynamical rigidity of Fano manifolds of Picard number 1 (with an appendix by Jie Liu)

Let \(f:X\rightarrow Y\) $f:X\to Y$ be a surjective morphism of Fano manifolds of Picard number 1 whose VMRTs at a general point are not dual defective. Suppose that the tangent bundle \(T_X\) $T_X$ is big. We show that \(f\) $f$ is an isomorphism unless \(Y\) $Y$ is a projective space. As applications, we explore the bigness of the tangent bundles of complete intersections, del Pezzo manifolds, and Mukai manifolds, as well as their dynamical rigidity.