Surface x(u, v) and curve α(t) on it given by u(t) & v(t). Math 4140/5530: Differential Geometry

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1 Surface x(u, v) and curve α(t) on it given by u(t) & v(t). α du dv (t) x u dt + x v dt

2 Surface x(u, v) and curve α(t) on it given by u(t) & v(t). α du dv (t) x u dt + x v dt ( ds dt )2

3 Surface x(u, v) and curve α(t) on it given by u(t) & v(t). α du dv (t) x u dt + x v dt ( ds dt )2 α (t) 2 α (t) α du (t) ( x u dt + x v dv dt ) ( x u du dt + x v dv dt )

4 Surface x(u, v) and curve α(t) on it given by u(t) & v(t). α du dv (t) x u dt + x v dt ( ds dt )2 α (t) 2 α (t) α (t) ( x u du dt + x v dv x u x u ( du dt )2 du + 2 x u x v dt dv dt + x v x v ( dv dt )2 dt ) ( x u du dt + x v dv dt )

5 Surface x(u, v) and curve α(t) on it given by u(t) & v(t). α du dv (t) x u dt + x v dt ( ds dt )2 α (t) 2 α (t) α (t) ( x u du dt + x v dv x u x u ( du dt )2 du + 2 x u x v dt E( du dt )2 + 2F du dt dv dt + G( dv dt )2 dv dt + x v x v ( dv dt )2 dt ) ( x u du dt + x v dv E, F, G play important roles in many intrinsic properties of a surface like length, area and angles Example 1: x(u, v) (u, v, 0) dt )

6 [ ] [ ] g11 g Matrix representation: g ij 12 E F g 21 g 22 F G g ij determines dot products of tangent vectors

7 [ ] [ ] g11 g Matrix representation: g ij 12 E F g 21 g 22 F G g ij determines dot products of tangent vectors E, F, G play important roles in many intrinsic properties of a surface like length, area and angles (ds ) 2 (du ) 2 du dv E + 2F dt dt dt dt + G( dv ) 2 dt ds 2 g 11 du 1 du 1 +g 12 du 1 du 2 +g 21 du 2 du 1 +g 22 du 2 du 2 i,j g ij du i du j

8 [ ] [ ] g11 g Matrix representation: g ij 12 E F g 21 g 22 F G g ij determines dot products of tangent vectors w 1, w 2 in T p M { x u, x v } is a basis: w 1 a x u + b x v, w 2 c x u + d x v w 1 w 2 foil

9 [ ] [ ] g11 g Matrix representation: g ij 12 E F g 21 g 22 F G g ij determines dot products of tangent vectors w 1, w 2 in T p M { x u, x v } is a basis: w 1 a x u + b x v, w 2 c x u + d x v foil w 1 w 2 ac x u x u + (ad + bc) x u x v + bd x v x v ace + (ad + bc)f + bdg

10 [ ] [ ] g11 g Matrix representation: g ij 12 E F g 21 g 22 F G g ij determines dot products of tangent vectors w 1, w 2 in T p M { x u, x v } is a basis: w 1 a x u + b x v, w 2 c x u + d x v foil w 1 w 2 ac x u x u + (ad + bc) x u x v + bd x v x v ace + (ad + bc)f + bdg a(ce + df) + b(cf + dg) [ a b ] [ ] ce + df cf + dg

11 [ ] [ ] g11 g Matrix representation: g ij 12 E F g 21 g 22 F G g ij determines dot products of tangent vectors w 1, w 2 in T p M { x u, x v } is a basis: w 1 a x u + b x v, w 2 c x u + d x v foil w 1 w 2 ac x u x u + (ad + bc) x u x v + bd x v x v ace + (ad + bc)f + bdg a(ce + df) + b(cf + dg) [ a b ] [ ] ce + df cf + dg [ a b ] [ ] [ ] E F c F G d

12 [ ] [ ] g11 g Matrix representation: g ij 12 E F g 21 g 22 F G g ij determines dot products of tangent vectors w 1, w 2 in T p M { x u, x v } is a basis: w 1 a x u + b x v, w 2 c x u + d x v foil w 1 w 2 ac x u x u + (ad + bc) x u x v + bd x v x v ace + (ad + bc)f + bdg a(ce + df) + b(cf + dg) [ a b ] [ ] ce + df cf + dg [ a b ] [ ] [ ] E F c F G d w 1 w 2 w 1 w 2 cos θ E, F, G play important roles in many intrinsic properties of a surface like length ( ds dt )2, area (det) and angles (above)

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14 2nd Fundamental Form l x uu U, m x uv U, n x vv U 2nd picture: The Center of Population of the United States curve: κ, τ rate of change of unit vector fields T & B ( N). surface: U unit vector field. Whole plane of directions rates of change of U are measured, not numerically, but by a linear operator called the shape operator, which captures the bending of a surface.

15 2nd Fundamental Form l x uu U, m x uv U, n x vv U 2nd picture: The Center of Population of the United States curve: κ, τ rate of change of unit vector fields T & B ( N). surface: U unit vector field. Whole plane of directions rates of change of U are measured, not numerically, but by a linear operator called the shape operator, which captures the bending of a surface. S p ( w) w U S( x u ) x u x uu U l, S( x u ) x v x uv U m, S( x v ) x v x vv U n eigenvalues of the shape operator: max and min normal curvature at p, called the principal curvatures κ 1 and κ 2