differential geometry w notes from teacher_Part_72

differential geometry w notes from teacher_Part_72 -...

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Unformatted text preview: Chapter 6 Connection and Curvature 6.1 Affine Connection 6.1.1 Covariant Derivative Definition 6.1.1 Let M be an n-dimensional manifold. An affine connection is an operator : C ∞ (T M ) × C ∞ (T M ) → C ∞ (T M ) that assigns to two vector fields X and Y a new vector field X Y, that is linear in both variables, that is, for any a, b ∈ R and any vector fields X, Y and Z, • + b Z) = a X Y + b aX+bY Z = a X Z + b Y Z X (aY XZ and satisfies the Leibnitz rule, that is, for any smooth function f ∈ C ∞ ( M ) and any two vector fields X and Y, X ( f Y) = X( f )Y + f X Y = (d f )(X)Y + f X Y • Let xµ , µ = 1, . . . , n, be local coordinates and ∂µ be the basis of vector fields. It will be called a coordinate frame for the tangent bundle. • A basis of vector fields ei = eµ ∂µ , i = 1, . . . , n, for the tangent bundle T M is i 143 144 CHAPTER 6. CONNECTION AND CURVATURE called a frame. • We will denote partial derivatives by ∂i f = f,i and the action of frame vector fields on functions by ei ( f ) = eµ ∂µ = f|i . i • For any frame the commutator of the frame vector fields defines the commutation coefficients [ei , e j ] = C k i j ek . • The commutation coefficients are scalar functions, in general. • Theorem 6.1.1 A frame ei is a coordinate frame if and only if for any i, j = 1, . . . , n, [ei , e j ] = 0 . Proof : 1. Need to show that there is a local coordinate system xi such that ei = ∂i . 2. Use the dual basis of 1-forms to prove that they are exact. Definition 6.1.2 Let {ei } be a frame of vector fields and σ j be the dual frame of 1-forms. The symbols ωi jk defined by • ωi k j = σ i ( e j ek ) are called the coefficients of the affine connection. • We denote i = ei . • Then ie j = ωk ji ek , diffgeom.tex; April 12, 2006; 17:59; p. 143 ...
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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