ArcGIS Server Java Service Packs occasionally do not deploy the required JAR files causing a problem in launching the ArcGIS Server Manager and/or problems in starting the ArcGIS Server Manager Service.
It appears that the solution for the time being will be to re-deploy SP3 (or SP2 if the problem was reported after deploying SP2) and verify that the arcgis_manager.jar file is copied under the $ARCGIS_HOME\java\manager\services\lib folder). The Post Install should be run afterwards and rebooting the machine after deploying SP3 (or SP2, as mentioned above) is recommended as well.