I'm interested to know what mathematical and physics topics Einstein learnt during his graduation.Did he know about advanced mathematical topics such as Hamiltonian mechanics ,Lagrangian mechanics, Manifold and Differential topology .
It is very difficult to access what he really knew from reading or from doing the things himself, but this paper address the simpler questions, what classes he took and what were his scientific contacts in the student years:
Let me also add that "manifolds" and "differential topology" were not formalized at that time, and these words were not used. On the other hand, some physicists and mathematicians has quite good understanding of these things. For example, the preliminary chapter of Maxwell's Treatease on electricity and magnetism shows that he had quite a good understanding of differential forms, homology and cohomology, Morse theory and Hodge theory (all in dimension 3, of course), the things which were formally invented or discovered at a much later time. There is little doubt that Einstein knew about this book, and it was easily available to him, but how much did he actually read or understand of it, probably cannot be determined.
EDIT. I actually did not read the paper referred above when I wrote this answer. Now I read it carefully, and it is indeed amazing:-) Einstein had Adolf Hurwiz as his math professor, one of the greatest mathematicians of that time. Another professor was Hermann Minkowski. But Einstein's classmates say he skipped their seminars. Instead, as a friend recalls:
Einstein felt that "the most fascinating subject at the time that I was a student was Maxwell's theory"
Somehow it happened that mathematicians of 19 and early 20 century did not read Maxwell's book (and of course they do not read it nowadays. Who cares about 19-th century books!) There is a very interesting paper:
Dyson, Freeman J. Missed opportunities. Bull. Amer. Math. Soc. 78 (1972), 635–652. http://www.ams.org/journals/bull/1972-78-05/S0002-9904-1972-12971-9/S0002-9904-1972-12971-9.pdf Especially pp. 638-9. My own experience confirms everything Dyson says there.
He discusses exactly this subject: what would mathematics gain if mathematicians read Maxwell. Maxwell actually anticipated much of the most important mathematics of the 20-th century.
It is amazing how mature Einstein was in his student years: he knew what to study:-) Instead of attending lectures, he studied himself "the works of Bolzmann, Kirchhoff, Helmholtz, Hertz etc.", according to his own recollection, and perhaps of Maxwell as well.
EDIT. I recently found a Russian translation of Einstein's paper "Electrodynamics of moving bodies", 1905 and read it. It is frequently mentioned that this paper has no reference list, and Einstein gives no credit to anyone. But Maxwell is mentioned frequently, (without exact references).
Arthur I. Miller's Albert Einstein’s Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905–1911) has on p. 81 a section "On Einstein's Knowledge of Electrodynamics in 1905".
I don't have access to it at the moment, but if I remember correctly, that section lists the books and monographs by Lorentz, Hertz, Föppl, Boltzmann, Helmholtz, Kirchhoff and others that Einstein had "definitely", "very probably", or "maybe" read by that time.
Edit 1: Having found the book, here goes (it's mostly electrodynamics, except Kirchhoff's book on mechanics):
Edit 2: Einstein’s Collected Papers reproduce his ETH Record and Grade Transcript which shows all the classes he took for credit. An Appendix has details of the course syllabi. (To the OP’s specific question, see Geiser’s courses on Infinitesimalgeometrie, and Minkowski’s on Analytische Mechanik + Anwendungen.)
Edit 3: There is also an entire book on the subject.
The following screenshots (linked in a prior answer) are taken directly from Volume 1 of "The Collected Papers of Albert Einstein" and are all the courses he was required to take during his undergraduate years at the Polytechnic of Zurich. The grades are also shown. As we can see, Einstein did not know about differential geometry, or to be more precise, about absolute differential calculus which was the theory of tensors developed by Ricci Curbastro, Levi Civita and others in the decades before. He will learn these topics in a second time, while working on General Relativity. He will have many correspondences with Tullio Levi-Civita, who helped him with the math as well as his good friend mathematician Grossmann.