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...Differential geometry rui loja fernandes contents preface part basic concepts lecture manifolds as subsets of euclidean space abstract with boundary partitions unity tangent and the dierential immersions submersions submanifolds embeddings whitney s theorem foliations quotients lie theory vector fields flows bracket derivative distributions frobenius groups algebras integrations exponential transformation forms tensor cartan calculus integration on de rham cohomology homotopy invariance mayer vietoris sequence computations in degree index fiber bundles thom class euler pull backs classication connections parallel transport curvature holonomy characteristic classes principal these are notes for courses dierentiable i ii that am lecturing at uiuc this course is usually taken by graduate students mathematics their rst or second year studies background a knowledge analysis algebra topology mymainaiminwritinguptheselecturesnotesistooerawrittenversion lectures should give chance to concentra...