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...Part iii dierential geometry based on lectures by j a ross notes taken dexter chua michaelmas these are not endorsed the lecturers and i have modied them often signicantly after they nowhere near accurate representations of what was actually lectured in particular all errors almost surely mine this course is intended as an introduction to modern it can be with view further studies topology should also suitable supplementary if your main interests for instance analysis or mathematical physics tentative syllabus follows local manifolds denition examples smooth maps tangent vectors vector elds bundle geometric consequences implicit function theorem submanifolds lie groups bundles structure group example hopf mor phisms automorphisms exterior algebra forms tensors symplectic orientability partitions unity integration stokes de rham cohomology derivative connections covariant derivatives curvature bianchi identity riemannian torsion s identities free metrics levi civita con nection christoe...