{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

differential geometry w notes from teacher_Part_72

# differential geometry w notes from teacher_Part_72 -...

This preview shows pages 1–2. Sign up to view the full content.

Chapter 6 Connection and Curvature 6.1 A ffi ne Connection 6.1.1 Covariant Derivative Definition 6.1.1 Let M be an n-dimensional manifold. An a ffi ne con- nection 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 , X ( a Y + b Z ) = a X Y + b X Z a X + b Y Z = a X Z + b Y Z 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 x μ , 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 e i = e μ i μ , i = 1 , . . . , n , for the tangent bundle T M is 143

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 e i ( f ) = e μ i μ = f | i . For any frame the commutator of the frame vector fields defines the
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern