Generalized Curvatures Part 3 pdf

3.2 The Pointwise Gauss Curvature 33

3.2 The Pointwise Gauss Curvature

After giving the basic definition of the Gauss curvature at a point of a smooth sur￾face or a vertex of a polyhedron, this section focusses on the difficulty of getting

pointwise approximations or convergence theorems.

3.2.1 Background on the Curvatures of Surfaces

Let S be a (smooth) oriented surface of the (oriented) Euclidean space E3. The sur￾face S can be endowed with the Riemannian metric induced by the canonical scalar

product of E3. Classically, the Gauss curvature function G of S can be defined with

the metric on S. In our context, we prefer to define it by using the extrinsic structure

of S. Theorema egregium of Gauss asserts that both definitions are equivalent.

We denote by T S the tangent bundle of S and by ξ the unit normal vector field of

S. If p is a point of S, S can be locally defined around m by a smooth immersion

x : U → E3,

where U is a domain of E2. Let (u, v) be a system of coordinates on U.

Since |ξ | = 1,

(

∂ ξ

∂u

)p and (

∂ ξ

∂ v

)p

are orthogonal to ξp and define a frame of TpS. Consequently, the differential Dξ

induces a tensor

A : T S → T S,

defined for each vector X ∈ T S by

A(X) = −DX ξ .

The tensor A is self-adjoint with respect to the scalar product <,>: for every

X,Y ∈ T S,

< A(X),Y >=< X,A(Y) > .

The tensor A is called the Weingarten tensor and its adjoint h, defined for every

X,Y ∈ T S by

h(X,Y) =< A(X),Y >,

is called the second fundamental form of S. At each point p of S, the eigenvalues

λ1p and λ2p of h at p are called the principal curvatures of S at p. At each point p,

there exist two (orthogonal) eigenvectors of A, called the principal directions of S

at p. The integral lines of these principal directions are called the lines of curvature

of S.

34 3 Motivation: Surfaces

Fig. 3.7 At the point p, the

two principal lines (tangent to

the principal directions) are

orthogonal

S

ξ

p

The determinant of h is the Gauss curvature Gp of S at p and half the trace Hp of

h is the mean curvature of S at p:

Gp = λ1pλ2p ,

Hp = 1

2 (λ1p +λ2p ).

Another (equivalent) point of view is to define the principal curvatures and direc￾tions as follows: a geodesic of S is a curve minimizing the distance locally. It is well

known that, at each point p of S and for any (unit) tangent direction X at p, there

exists a unique (local) geodesic γX tangent to X. When X varies on the unit circle

S1 ⊂ TpS, the curvature kp(X) at p of γX (considered as a curve in E3) is related to

the second fundamental form as follows:

kp(X) = hp(X,X).

The maximum and minimum of kp(X) are the two principal curvatures λ1p and λ2p

of S at p. An extensive study of the local Riemannian geometry of surfaces in E3

can be found in [40] (Fig. 3.7).

3.2.2 Gauss Curvature and Geodesic Triangles

Let us present a discrete point of view: it is possible to recover the Gauss curvature

at a point p of a smooth surface S with help of the solid angle of geodesic triangles

incident to p (see [2] for a extensive study). If v,v1,v2 are three points on S, the

geodesic triangle with vertices v, v1,v2 is the union of the geodesic arc γ1 joining

v and v1, the geodesic arc γ2 joining v and v2, and the geodesic arc γ3 joining v1

and v2. If τ is a geodesic triangle on S, with vertices v,v1,v2, denote by l,l1,l2 the

lengths of the geodesic arcs opposite to the vertices v,v1,v2. Finally, let α be the

angle of τ at v.

From this geodesic triangle, we construct now an Euclidean triangle t whose

vertices are p, p1, p2 in E2 and whose edge lengths are

l = |p1 p2|,l1 = |pp2|,l2 = |pp1|.

Let β be the angle of t at p. A fundamental result can be stated as follows.