Is it possible to realize a homotopy between two vector bundles (which are sub-bundles of the tangent bundle) over the same base $B$ as induced by an ambient isotpy? In other words, assume $X_t $ is a continuous family of sub-bundles of $TB$ of the same rank, connecting two sub-bundles $X_0$ and $X_1$ over base $B$. Is it possible to find an isotopy $\phi_t:M \to M $ so that $(\phi_t)_\star X_0=X_t$?
2026-03-28 01:47:19.1774662439
homotopy via isotopy
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