If you move within Germany, are not receiving benefits and your residence title does not contain any restrictions such as a residence requirement, you can simply move and do not have to inform the Leipzig authorities.
The only thing that have to remember is that in Germany you must report to the new Residents’ Registration Office (Einwohnermeldeamt) within two weeks. You can do this at any Citizens Advice Bureau in your new town/city of residence, you can show your new rental contract and accommodation provider confirmation along with your ID, and then you are registered in your new town/city.
However, if you are in receipt of benefits and leave the jurisdiction of the authority you are receiving those benefits from, you must apply to the relevant authority (e.g. Job Centre or Social Welfare Office) for relocation and re-registration. Currently, this can be done by sending an informal email to the relevant case worker. The authority in your new town/city of resident will then send the essential information to the authority in your former town/city. This simplified process will probably change soon, so it is always a good idea to discuss any intention to move with your case worker in person.