differential geometry w notes from teacher_Part_65

# differential geometry w notes from teacher_Part_65 - 129...

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

5.4. DEGREE OF A MAP 129 Then N is a vector field that is everywhere normal to M . Notice that the norm of the normal vector is || N || 2 = || e 1 || 2 || e 2 || 2 - ( e 1 , e 2 ) 2 . The second fundamental form , or the extrinsic curvature is defined by the matrix b μν = 1 || N || e μ u ν , N = 1 || N || 2 x i u μ u ν N i . The second fundamental form describes the extrinsic geometry of the sur- face M . The mean curvature of M is defined by H = g μν b μν . The Gauss curvature of M is defined by K = det b μν det g αβ . Gauss has shown that the K is an intrinsic invariant. In fact, K = R 12 12 = 1 2 R , where R 12 12 is the only non-vanishing components of the Riemann curvature of the metric g and R is the scalar curvature. This will be discussed later. The Gauss map is the map ϕ : M S 2 from M to S 2 defined by ϕ ( x ) = N ( x ) || N ( x ) || , that is, it associates to every point x in M the unit normal vector at that point.

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