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We propose a conjectural list of Fano manifolds of Picard number $1$ withpseudoeffective normalized tangent bundles, which we prove in varioussituations by relating it to the complete divisibility conjecture of Russo andZak on varieties with small codegree. Furthermore, the pseudoeffectivethresholds and hence the pseudoeffective cones of the projectivized tangentbundles of rational homogeneous spaces of Picard number $1$ are explicitlydetermined by studying the total dual VMRT and the geometry of stratified Mukaiflops. As a by-product, we obtain sharp vanishing theorems on the globaltwisted symmetric holomorphic vector fields on rational homogeneous spaces ofPicard number $1$.

Normalized tangent bundle, varieties with small codegree and pseudoeffective threshold