HW3 070909

# HW3 070909 - Differential Geometry Homework 3 26.05.08 Hava...

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

Differential Geometry Homework 3 - 26.05.08 Hava Shabtai, ID 043039619, Department of Mathematics, University of Haifa Email: —[email protected] 1. Was proven in the previous home work. 2. Was proven in the previous home work. 3. We can use one-to-one correspondence in order to get an alternative definition of the Tangent Space. In that definition, for a smooth manifold X and a point x in X , the Tangent Space T x ( X ) is the set of all derivation of C ( X ) at x , where a linear map, F : C ( X ) -→ R is called a derivation at x if it satisfies F ( gh ) = g ( x ) F ( g ) + h ( x ) F ( h ) for all g, h C ( X ) . Now let ϕ : U X -→ D be a chart containing x (we toke ϕ := φ - 1 the inverse of the chart from the notion in class, since it is a diffeomorphism it is well defined, and it will be a great use in this question and the next) . Let us write a new sign ∂x i x that acts on a smooth function f : U -→ R by : ∂x i x f := ∂x i ( f ϕ - 1 ) ϕ ( x ) .

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

View Full Document
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