CompConsider a smooth $\omega$--periodic differential system in  
${\bf R} \times {\bf R}^{n}$, say $S _\omega$, of ordinary differential equations, and let $E$
be an equilibrium for $S _\omega$. Preliminarily it is shown that the total stability of $E$ is 
equivalent to the existence of a fundamental family of asymptotically stable 
neighborhoods of $E$. Thus a known theorem of Seibert   \cite {S} concerning autonomous 
systems is extended to periodic systems. Let us assume now the existence of a smooth 
invariant manifold $\Phi$  in ${\bf R} \times {\bf R}^{n}$, containing ${\bf R} \times  \{E\}$, 
$\omega$--periodic in $t$, and asymptotically stable ``near'' $E$.  By using the above 
extension of  Seibert's  theorem and some previous results in our paper  \cite {SV1}, 
  \cite {SV2}, we prove here that if $E$ is totally stable on  $\Phi$ (that is with respect to the 
solutions lying on  $\Phi$), then $E$ is unconditionally totally stable.