Chapter 5 Geometry of regular surfaces Flashcards by Dominique Gabriela

intrinsic geometry

intrinsic geometry, that is the study of geometric properties
that are seen by the inhabitants of a surface only.

e.g if we bend a surface the length of curve wont change from point of observers geometry wont change but the shape of surface changes

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dot product

in R^n is a positive definite symmetric bilinear form on R^n

dot prod refers to notion of chosen basis
for any scalar product B on R^n there exists basis
e_1‾,….,e_n‾ s,t for any vectors u and v in R^n

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symmetric bilinear form,

Let V be a finite-dimensional vector space. A map B : V × V → R is called a symmetric bilinear form, if it satisfies the following properties

(i) B(α1u1 + α2u2, v) = α1B(u__1, v) + α2B(u2, v) for any vectors u1, u2, and v ∈ V , and any real
numbers α1 and α2;

(ii) B(u, v) = B(v, u) for any vectors u and v ∈ V

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positive definite

A symmetric bilinear form B on a vector space V is called positive definite (or a scalar product), if B(u, u) > 0 for any u ∈ V , and B(u, u) = 0 if and only if u = 0

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matrix of the basis

by property 2: symmetric b_ij=b_ji for any i and j

A MATRIX CAN REPRESENT BILINEAR FORMS

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First Fundamental Form G

(RESTRICTION to the tangent space of the dot prod from euclidean spae R^n)

on a regular surface Σ^n ⊂ R^m is a family of scalar products G_p : T_pΣ^n×T_pΣ^n → R (that is a family of positive definite symmetric bilinear forms), where

p ranges over Σ^n, such that
G_p(X, Y ) = X · Y for any X, Y ∈ T_pΣ^n and any p ∈ Σ^n
——————————–

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Let P_n be space of polynomials
f:R to R whose degree not greater than n
is this a positive definite symmetric bilinear form on P_n:
B(f,g) =∫_{0,1} f(t)g(t).dt

We claim that the scalar product B is
uniquely determined by the matrix bij = B(ei
, ej ), where i, j = 1, . . . , n. Indeed, for any vectors u and v ∈ V by properties (i) and (ii)

B(u,v) = sum_i,j b_ij u_i v_j
Thus, we conclude that the value B(u, v) can be computed from the knowledge of the matrix (bij ), and the claim holds.

checking linear wrt first var
non neg integral is non neg

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measuring length of vector in Bilinear form

distance and angles

given vector in V
length in sense of given bilinear form

they can also imply cs
b(u,v) <= sqrt (B(u,u) B(v,v))

so we can do everything that we can in euclidean space

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CS bilinear form

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bilinear triangle inequality

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interested inpath length?
We are concerned with computing
the lengths of curves on surfaces Σ^n ⊂ R^m

Let γ : [a, b] → R^m be a PC such that γ(t) ∈ Σ
n for any t ∈ [a, b]. As we know, the length of γ is given by the formula
L(γ) = integral_[a,b] |γ’(t)| dt.

Since γ lies in Σn, the velocity vector γ’(t) lies in the tangent space Tγ(t)Σ^n; due Definition IV.15
(to show γ’(t) lies in tangent space sufficient to construct generating curve γ~ :(-e,e) to surface s,t γ~(0)=p and γ~’(0)=γ’(t)

we are interested in the geometry of the surface ie being able to compute lengths of paths that lie in surface

and hence we only need to know values |X| for |X| in the tangent space for all p in the surface
(speeds lengths for tangent vectors AND THATS ALL WE NEED TO KNOW)

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Consider the space P_2 of polynomials f : R → R whose degree is not greater than two. Pick a basis in this space and compute the matrix of the form given by formula corresponding to it

what is dimension of this space?

what is dimension of this space?
basis consists of 3 polynomials so it is 3
basis
(1,t,t^2)
compute
matrix of
B(f,g) = integral_[0,a] f(t)g(t).dt in this basis

bij = B(ei, ej )
integrating
1, t, t^2
t, t^2, t^3
t^2, t^3, t^4

integrating over [0,1]

[ 1 1/2 1/3]
[1/2 1/3 1/4]
[1/3 1/4 1/5]

on 6 components

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MATRICES AND COMPUTATION

for some chart (U,phi,V) around p in Σ^n
we have a CANONICAL BASIS
X_i(p) = D_x(ϕ(e_i)) = ∂ϕ/∂xi(x),

of tangent space and we may compute corresponding matrix in this chart

we obtain
gij (ϕ(x)) = G_ϕ(x) (Xi, Xj ) = ∂ϕ/∂xi(x) ·∂ϕ/∂xj
(x)

if p lies in this chart then
D_x(ϕ) :R^n to tangent space
is a linear isometry
in maps the standard basis in R^n to the tangent space spanned by above

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THE FIRST FUNDAMENTAL FORM

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for a given chart the first fundamental form

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G_p for a given point on surface and basis

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Using the First Fundamental Form we can now re-write the formula for the length of a curve on
Σn in the following fashion:

L(γ) = integral_[a,b]
SQRT(Gγ(t)(γ’(t), γ’(t))dt.

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so how would we compute the arc length? of a curve

if you are inhabitant of the surface and you dont know the formula, directions and lengths of tose directions not tangent to surface

gij (ϕ(x)) = G_ϕ(x) (Xi, Xj ) = ∂ϕ/∂xi(x) ·∂ϕ/∂xj
(x) we know

could write coordinates on surface
ϕ-1 . γ(t) = (γ¹(t),…, γⁿ(t))

speed of tangent vector: in terms of first fundamental form speed would be
|γ’(t)|_R^n = sqrt(G_γ(t)( γ’(t), γ’(t))

so using this into integral of:
sqrt[sum_ij (gij(γ¹(t),…,γⁿ(t))) dγ^i/dt (t)dγ^j/dt(t) ].dt

FORMULA FOR THE LENGTH FOR __SURFACE
has no idea what R^n is

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Let Σn ⊂ R^{n+1} be a graph of a smooth function f : Ω → R, that is the set
Σ^n = {(x, f(x)) : x ∈ Ω}.

using charts

we have only one chart (Ω, ϕ, Ω×R), where the map
ϕ : Ω → Ω×R is given by
ϕ(x) = (x, f(x)).
(cylinder Ω×R

Using relations
X_i(p) = D_x(ϕ(e_i)) = ∂ϕ/∂xi(x),
=(0,…,1,0,..,0,∂f/∂x_i), for ith place
we obtain the following formula for the matrix of the first fundamental form:
g_ij (ϕ(x)) =
∂ϕ/∂xi(x) ·∂ϕ/∂xj(x)
=X_i . X_i
= δ_ij +∂f/∂xi(x) ∂f/∂xj(x),
where i, j = 1, . . . , n, and
δij is the Kronecker delta. δij = 1 iff i = j,

in this e.g if f(x) =c constant the surface is just the plane itself, so becomes identity matrix as the euclidean matrix

e.g f(x,y)=x^3+y^3 decide on curve (t,0)
inhabitants see, but if we live in R^3 for us the curv will be gamma(t)=(t,0, t^3 +0)
g_ij (ϕ(x)) =
= δ_ij +∂f/∂xi(x) ∂f/∂xj(x),
…9x^2 etc

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matrix representation

The family of matrices (gij ) ◦ ϕ defined by relation (V.7) is called the matrix
representation of the first fundamental form in a given coordinate chart (U, ϕ, V ) on a regular surface Σ^n ⊂ R^m.

(not discussed)

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e.g
Unit sphere S^n ⊂ R^{n+1). Consider a unit sphere S^n ⊂ R^{n+1}

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Hyperplace subset viewed as graph of 0 function

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why is first fundamental form useful?

INNER GEOMETRIC

geometric properties/quantaties of a regular surface that are determined by the first fundamental form only are called intrinsic or INNER GEOMETRIC

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INTRINSIC DISTANCE FUNCTION

an example of an intrinsic quality

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example instrinsic distance function on unit sphere **SPHERICAL DISTANCE**

PROP 5.1 **inequality for intrinsic distance**

it has constant speed then using CS inequality and corollary 2.22

example: on a plane the instrisic distance coincides with euclidean distance

the inequality might be strict only for a curved shape

LOCAL ISOMETRY BETWEEN SURFACES

when is a local isometry a GLOBAL ISOMETRY

A local isometry is called a global isometry if it is a bijective map between surfaces

REMARKS ABOUT ISOMETRIES relationships first fundamental forms

example for an ORTHOGONAL TRANSFORMATION map restricted is a GLOBAL ISOMETRY OF THE UNIT SPHERE

example: PLANE AND CYLINDER REGULAR 2 DIM SURFACES SHOW THERE IS A LOCAL ISOMETRY BETWEEN THE TWO

INTRINSIC QUALITY AND GLOBAL ISOMETRY

Using the notion of isometry, one can re-state Definition V.9 in the following form: a geometric quantity is called intrinsic if it is ”preserved” by all global isometries.

PROPN 5.2 local isometry instrinsic distance functions inequality

corollary 5.3 GLOBAL ISOMETRY and intrinsic distance

COROLLARY 5.4 **global isometry** **Euclidean isometry** did he go over these?

Corollary V.4. Let Φ : R^n → R^n be a smooth map. Then Φ is a **global isometry** of R n in the sense of Definition V.13 if and only if it is a **Euclidean isometry** in the sense of Definition I.20.

hypersurface

regular surfaces whose dimension is one less than the dimension of the ambient space

**GAUSS MAP** Is applied to a hyperspace units nomal field

diagram: VECTORS coming out at different points on surface p N_p and q N_q these form our unit normal field

EXAMPLE of **GAUSS MAP** of the unit sphere

further on example of S^n in R^{n+1) sphere

given this surface this example will have tangent space thus giving the normal fields on p with different orientations N(p) =p is a unit normal field the Gauss map coincides with identity N(p)-p is another unit normal field corresponding gauss map is called antipodal map

**SHAPE OPERATOR**

**propn shape operator is self adjoint**

eigenvalues always real

self-adjoint linear operator

**SHAPE OPERATOR PROPERTIES** PRINCIPLE CURVATURES MEAN CURVATURES GAUSS CURVATURES

UNIT SPHERE shape operator principle curvatures mean gauss curvature

differwntial N_1 =-id as k_n(p)=-1 D_p N_2=id

HYPERPLANE shape operator principle curvatures mean gauss curvature

Can you see that principal curvatures in Definition V.19 depend on an orientation on Σn? More precisely, can you see that principal curvatures change sign when a unit normal field N is replaced by −N? reversed orientation changes sign of principle curvatures

**SECOND FUNDAMENTAL FORM**

COROLLARY 5.6 second fundamental form for tangent vectors XY for a regular hypersurface oriented by unit normal N

COROLLARY 5.7 proving the PROP 5.5

PROPN 5.8 relationship between (b_ij) gauss curvature and g_ij useful?

SUMMARY properties of sec fund form

applying our useful relations: EXAMPLE cylinder

CYLINDER EXAMPLE notes

cylinder example working out partial derivs

spent all lecture...

gauss curvature: plane and cylinder locally isometric their curvatures related?

instrinsic? curvatures?

the principal curvatures, mean curvature (and hence, the shape operator and second fundamental form) are not intrinsic geometric quantities. GAUSS CURVATURE IS

Theorem V.9 (Theorema Egregium)

summary

EXAMPLE: using Theorem V.9 (Theorema Egregium) on eucludean plane and sphere

thm gauss curvature vanishes then

rotation index visually: with signed curvature

L5: missing recording lecture

COVARIANT DERIVATIVE AS INTRINSIC DERIVATIVE SECTION

l5: PRE SECTION I MISSED

L5 prop 5.1:

**LIE BRACKET** and properties

IT DOESNT DEPEND ON GEOMETRY OF SURFACE AND IF IS DIFFEOMORPHISM...

LIE BRACKET properties

lie bracket example from lecture

**C(surface^n)**

**CURVATURE TENSOR) level 5**

measures how far the separation

**CURVATURE TENSOR PROPERTIES** prop 5.2

e.g multiplied by function factors out.... R is an intrinsic geometric quantity

corollary 5.3 another curvature tensor

missed

remark

for Euclidean spaces the curvature tensor vanishes identically. Computation of the curvature tensor for general surfaces can be a highly non-trivial problem. Later we will see that for two-dimensional surfaces in R^ 3 the curvature tensor is entirely determined by the Gauss curvature

example: curvature tensor for n dimensional euclidean space covered by a chart

missed