I've read this about ABAP in Eclipse:
"The general idea of ABAP in Eclipse is to provide one Eclipse client that is installed on the developer PC and can connect to several backend systems from different releases. The connection is set up using a RFC/REST based protocol. The client provides a set of standard development tools such as the project explorer for system browsing and search views. In addition editors are available for each development object – classes, reports, function modules and so on. The development paradigm is still server based. This means that the development objects are stored solely on the backend (no local copies through check-in/check-out), and services such as syntax-check, search, where-used etc. run on the backend." (emphasis mine)
On the other hand Eclipse can Git in the current version (Oxygen.2 Release(4.7.2)).
How can I create local repositories from the ABAP objects I create / change in Eclipse with the ADT?