How to get a reference to LocationManager inside a Fragment

inside your fragment simply call this:

LocationManager mgr = 
(LocationManager)getActivity().getSystemService(Context.LOCATION_SERVICE);

So you simply get the Activity and call getSystemService() from there

Edit: As the getActivity method is deprecated since API 28 you can just use:

LocationManager mgr = 
    (LocationManager)getContext().getSystemService(Context.LOCATION_SERVICE);