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Let $f\colon X\to Y$ be a surjective morphism of Fano manifolds of Picardnumber 1 whose VMRTs at a general point are not dual defective. Suppose thatthe tangent bundle $T_X$ is big. We show that $f$ is an isomorphism unless $Y$is a projective space. As applications, we study the bigness of the tangentbundles of complete intersections, del Pezzo manifolds, and Mukai manifolds, aswell as their dynamical rigidity.

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