I suggest you have a look at this page : https://www.campusfrance.org/en/how-to-enrol-Doctorate-France and then specific universities pages.
I recommend you act swiftly as the process tends to start much earlier in the year, several offers have now been filled, but there are probably still some good opportunities if your search is wide enough.
Edit : here are some more details. On the following page you will see the results of the Masters students that have been accepted to a PhD position in mathematics at the University of Grenoble, which is strong in differential geometry. https://www.adum.fr/as/ed/page.pl?site=edmstii&page=alloc_modalite
As you can see :
A) the deadline to apply was in june
B) the results show that all but one students whogot accepted had done a Masters in France. This is not surprising since in mathematics most PhD positions are reserved for this case, i.e. this natiowide process where applicants with a french Masters are numerous and usually take up most of the positions since coming from a french Masters allows professors to get the grades of these students and the opinions of the colleagues who thaught them during the Masters. It's difficult to judge the level of an international student unless their candidacy has strong backing from a researcher whom the french ones know.
So to achieve your goal you'd have more chance to do a Masters in France next year, try to be among the top students, and apply. To succeed this year your Masters thesis should be really good, and you should have a recommendation letter from a good researcher, and finally you should try to see if a university, or a researcher with a special kind of grant, has some unfilled positions.