There is a natrual connection on the tangent bundle?

Question

I come up with a (maybe stupid) question: let $M$ be a smooth manifold, then the exterior differential $d$ is a natural connection on $\Omega^k(M)$, hence by dualizing we get a natural connection on $TM$? How can it be true (without a metric)?

Answer 1

Since you say "dualizing", I assume you're talking about the case $k=1$. If we view differential forms as alternating tensors, the exterior derivative $\Omega^1(M) \to \Omega^2(M)$ can be viewed as a map $\Gamma(T^*M) \to \Gamma(T^*M \otimes T^*M),$ which has the right "type signature" to be a connection; but it does not satisfy the (correct kind of) Leibniz rule: for a smooth function $f$ and a one-form $\theta$ we have $$d(f \theta) = f d \theta + df \wedge \theta \ne f d \theta + df \otimes \theta.$$