You can use ViewPager with native fragments from the android.app package with the adapters from the android.support.v13.app package. You have to use the v13 support jar for that.
There are two versions of the adapters that work with ViewPager, the ones in the v4 package are meant to be used with support fragments, the ones in v13 with native fragments.
The reason why there are now two fragment implementations is historical: Fragments in the android.app package were introduced with Android 3 for tablets only and the support library was created to bring fragments to phones running older versions. On Android 4 you have both.
From my own experience I would recommend using support fragments, even when developing for Android 4. Here are some reasons: Fragment or Support Fragment?