Riemannian Geometry and Geometric Analysis
From the reviews: 'This book provides a very readable introduction to Riemannian geometry and geometric analysis. The author focuses on using analytic methods in the study of some fundamental theorems in Riemannian geometry, e.g., the Hodge theorem, the Rauch comparison theorem, the Lyusternik and Fet theorem and the existence of harmonic mappings. With the vast development of the mathematical subject of geometric analysis, the present textbook is most welcome. It is a good introduction to Riemannian geometry. The book is made more interesting by the perspectives in various sections. where the author mentions the history and development of the material and provides the reader with references.' Math. Reviews. The 2nd ed. includes new material on Ginzburg-Landau, Seibert-Witten functionals, spin geometry, Dirac operators.
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De Rham Cohomology and Harmonic Differential
Parallel Transport Connections and Covariant
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algebra apply assume basis boundary bounded called chart choose closed compact complex compute conformal connection consider constant contained continuous converges coordinates Corollary critical points curvature curve defined Definition denote depends derivative determined diffeomorphism differentiable differentiable manifold dimension elements energy equations equivalent Euclidean example exists fiber finite follows formula function geodesic geometry given harmonic map hence holds holomorphic homotopic implies induces invariant isometry Jacobi field Lemma length linear Math means metric minimal Namely neighborhood obtain operator oriented orthonormal particular positive Proof result Riemannian manifold satisfies scalar sectional curvature sequence smooth SO(n solution space structure surface symmetric tangent tangent vector tensor Theorem theory topology transformation unique values vanishes variation vector bundle vector field write yields