That was the strange thing. Nothing happend after I connected a keyboard or a mouse to it and pressed any key or wiggled the mouse.
I´ll try to install the XScreenSaver and give feedback afterwards.
Thanks for your help @all!!
P.S.: After half an hour it seems to work. Final check tomorrow, Cheers.
P.P.S.: After a night the screen is still on. Thanks for your help. The problem was solved by installing the XScreenSaver and put the pi in “no screen saver mode”.