到凝乳 发表于 2025-3-21 17:53:35
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http://reply.papertrans.cn/84/8304/830307/830307_2.png和谐 发表于 2025-3-22 03:12:01
Curvature,e . is a vector field such that . = .. We already met in 2.64 the second covariant derivative of a function, which is a symmetric 2-tensor. This property is no more true for the second derivative of a tensor. However, . only depends on ..obtuse 发表于 2025-3-22 08:09:01
Analysis on Manifolds and the Ricci Curvature,ignore mathematical beings which locally behave like domains on ., just as manifolds locally behave like .. On the other hand, when doing Analysis on manifolds, it may useful to cut them into small pieces (cf. for example 4.65 and 4.68 below). These pieces are no more manifolds, but they will be manOpponent 发表于 2025-3-22 09:14:19
Universitexthttp://image.papertrans.cn/r/image/830307.jpg油毡 发表于 2025-3-22 13:42:14
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Springer-Verlag Berlin Heidelberg 1987disciplined 发表于 2025-3-22 23:27:50
http://reply.papertrans.cn/84/8304/830307/830307_8.pngsacrum 发表于 2025-3-23 02:52:43
Differential Manifolds,A subset . ⊂ . is an . . . if, for any χ ∈ ., there exists a neighborhood . of χ in . and a . submersion .: . → . such that . ⊓ . = . (0) (we recall tnat . is a submersion if its differential map is surjective at each point).Facilities 发表于 2025-3-23 09:04:13
Riemannian Metrics,A Riemannian metric on a manifold M is a family of scalar products defined on each tangent space . and depending smoothly on .: