Ivancevic_Applied-Diff-Geom

288 Applied Differential Geometry: A Modern Introductionfor unique functions f ia on U ⊂ M, so thatφ ∗ = e i ⊗ φ ∗ η i = f ia e i ⊗ ω a .Note that φ ∗ is a section of the vector bundle φ −1 T N ⊗ T ∗ M.The covariant differential operators are represented as∇ M X a = ω ab ⊗ X b , ∇ N Y i = η ij ⊗ Y j , ∇ ∗ ω a = −ω ca ⊗ ω c ,where ∇ ∗ is the dual connection on the cotangent bundle T ∗ M.Furthermore, the induced connection ∇ φ on E is∇ φ e i = ( η ij (Y k ) ◦ φ ) e j ⊗ f ka ω a .The components of the Ricci tensor and scalar curvature are definedrespectively byR M ab = R M acbc and R M = R M aa.Given a function f : M → , there exist unique functions f cb = f bc such thatdf c − f b ω cb = f cb ω b , (3.138)where f c = df(X c ) for a local orthonormal frame {X c } m c=1. To prove thiswe take the exterior derivative of df = ∑ mc=1 f cω c and using structureequations, we have0 = [df c ∧ ω c + f bc ω b ∧ ω bc ] = [(df c − f b ω cb ) ∧ ω c ] .Hence by Cartan’s lemma (cf. [Willmore (1993)]), there exist unique functionsf cb = f bc such thatdf c − f b ω cb = f cb ω b .The Laplacian of a function f on M is given by∆f = − Tr(∇df),that is, negative of the usual Laplacian on functions.