10) The Gauss Bonnet Theorem

Question 1

Prove

Answer 1

Question 2

Prove Hopf’s Umlaufsatz (Rotation theorem)

Answer 2

Question 3

Define a curvilinear polygon

Answer 3

Question 4

For a curvilinear polygon

Answer 4

Question 5

Prove

Answer 5

(6) is simply first line with double integral on LHS and integral of κ_g on RHS
(7) is that integral of θ dot is 2π - sum of δ_i

Question 6

Prove

Answer 6

(8) is that first time of LHS is equal to 2π

Question 7

Give equation? Prove?

Answer 7

Question 8

2 surface patches are compatible if

Answer 8

Question 9

An atlas for S, a subset of R^3, is?
How does this relate to a global surface?

Answer 9

Question 10

How are tori denoted?

Answer 10

T_g where g is the genus (number of holes)

Question 11

For any T_g with non-neg g?

Answer 11

Question 12

Define a triangulation of a global surface S

Answer 12

Question 13

Relate compact global surfaces to polygons

Answer 13

Every compact global surface has a triangulation with finitely many polygons

Question 14

Euler number χ of a triangulation of a compact surface

Answer 14

Question 15

Relate the triangulation of S, a compact global surface in R^3, to the surface area element

Answer 15

Question 16

Prove

Answer 16

Question 17

Apply //KdA =2πχ to unit sphere

Answer 17

The Euler number remains the same for any deformation that doesn’t tear when deforming

Question 18

The Euler number of the compact surface T_g is

Answer 18

2-2g

Question 19

Prove

Answer 19

Combine this with Euler number of compact surface T_g is 2-2g