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

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

View Full Document Right Arrow Icon
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 ) .
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern