Nonexistence of Almost Complex Structures on the product $S^{2m}\times M$

In this note we give a necessary condition for having an almost complex structure on the product $S^{2m}\times M$, where $M$ is a connected orientable closed manifold. We show that if the Euler characteristic $\chi (M)\mathrm{\ne }0$, then except for finitely many values of $m$, we do not have almost complex structure on $S^{2m}\times M$. In the particular case when $M = $\mathbb{C}$\mathbb P^n, n \neq 1$,weshowthatif$n \not \equiv 3 \pmod 4$then$S^{2m} \times \mathbb C \mathbb P^{n}$ has an almost complex structure if and only if $m=1,3$. As an application we obtain conditions on the nonexistence of almost complex structure on Dold manifolds.