## Differential Geometry |

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affine connection affine transformations Applying assume basis called canonical Chapter chart closed compact complex complex manifold complex structure components condition consider constant coordinate system corresponding covariant covering curvature curvature tensor curve defined definition denote derivative determined differentiable dimension direct element equations Euclidean Example Exercise exists expression frame function fundamental Gauss geodesic given gives Hence holomorphic homogeneous identity implies induced integrable invariant isometry isomorphism Kaehlerian manifold Lemma Lie algebra Lie group linear linear connection locally matrix metric g n-dimensional neighborhood normal Note obtain operator orthonormal parallel positive preceding section projection Proof Proposition Prove relation representation resp respectively restriction Riemannian manifold satisfies Show subgroup submanifold tangent tensor field Theorem tion transformation unique vanishes vector field vector space