Orientation and Orientability

Orientation and orientability

1. Orientation on a vector space Throughout this section let V be a vector space over R of ﬁnite dimension 0 0 0 n ≥ 1. For two bases B = b1, . . . , bn and B = b1, . . . , bn of V , the transition matrix n TB0,B = (aij)i,j=1 from B to B0 is an n × n matrix where n X 0 bi = ajibj. j=1 The following proposition records some basic facts about transition ma- trices: Proposition 1.1. The following hold:

(1) For any basis B of V we have TB,B = In, the n × n identity matrix. 0 00 (2) For any bases B, B , B of V we have TB00,B0 TB0,B = TB00,B. 0 −1 (3) For any bases B, B of V we have TB,B0 = TB0,B; in particular, det TB0,B 6= 0. Proof. We will give a proof of part (2) which is the central point of this proposition. Denote (aij)ij = TB,B0 and (dkl)kl = TB0,B00 . Thus for every i = 1, . . . , n we have n X 0 bi = ajibj j=1 and for every j = 1, . . . , n we have n 0 X 00 bj = dkjbk. k=1 Therefore n n n X 0 X X 00 bi = ajibj = ajidkjbk = j=1 j=1 k=1 n  n  X X 00 00  dkjaji bk = (TB00,B0 TB0,B)kibk k=1 j=1 and therefore TB00,B0 TB0,B = TB00,B In view of the above proposition, the following deﬁnition is natural: Deﬁnition 1.2 (Bases with the same orientation). Let B and B0 be bases of V . We say that B0 has the same orientation as B, and write B0 ∼ B, if 0 0 det TB0,B > 0. We B has the opposite orientation from B, and write B 6∼ B, if det TB0,B < 0. 1 2

We have: Proposition 1.3. The following hold: (1) The relation ∼ is an equivalence relation on the set B of all bases of V , that is: (a) For every basis B of V , B ∼ B. (b) If B0 ∼ B then B ∼ B0. (c) If B0 ∼ B and B00 ∼ B0 then B00 ∼ B. 0 0 (2) For any basis B = b1, . . . , bn of V we have B 6∼ B , where B = −b1, b2 . . . , bn. (3) There are exactly two distinct equivalence classes for the equivalence relation ∼ on B. Definition 1.4 (Orientation). An orientation on V is a choice of an equivalence class for an equivalence relation ∼ on the set B of all bases of V . In practice, an orientation on V is specified by choosing a specific basis B = b1, . . . , bn and taking the orientation to be the ∼-equivalence class of B. Then for any other basis B0 of V , the basis B0 is positively oriented with 0 respect to this orientation if det TB0,B > 0, and B is negatively oriented with respect to this orientation if det TB0,B < 0. n Definition 1.5 (Standard orientation on R ). For n ≥ 1 the standard ori- n n entation on R is determined by the standard unit basis e1, . . . , en of R . We record the following basic facts about the standard orientation in dimensions 2 and 3: Proposition 1.6. The following hold: 2 (1) A basis B = b1, b2 of R is positively oriented with respect to the 2 standard orientation on R if and only if moving from b1 to b2 is along the angle < π between these two vectors gives the movement in the counter-clockwise direction. 3 (2) An orthonormal basis B = b1, b2, b3 of R is positively oriented with 3 respect to the standard orientation on R if and only if b3 = b1 × b2. 3 (3) A basis B = b1, b2, b3 of R is positively oriented with respect to the 3 standard orientation on R if and only if (b1 × b2) · b3 > 0.

2. Orientable surfaces 2 Recall that we denote the coordinates on R by u, v and we denote the 3 coordinates on R by (x, y, z). 3 Deﬁnition 2.1. A surface S ⊆ R is said to be orientable if there exists a 2 collection of coordinate patches {ψα : Uα → S}α∈A (where each Uα ⊆ R is an open subset and ψα is an injective regular map) such that: (1) We have ∪α∈Aψα(Uα) = S. 3

(2) For every α, β ∈ A such that ψα(Uα) ∩ ψβ(Uβ) 6= ∅, we have −1 det D(ψβ ◦ ψα) > 0 −1 −1 at every point where ψβ ◦ ψα is defined (here D(ψβ ◦ ψα) is the −1 2 × 2 Jacobi matrix of the map ψβ ◦ ψα ). A collection {ψα : Uα → S}α∈A as above is called an orienting atlas for S. An orienting atlas {ψα : Uα → S}α∈A defines an orientation on TqS for every q ∈ S as follows: If q = ψα(p) for some α ∈ A and p ∈ Uα, we declare ∂ψα ∂ψα the basis (ψα)∗((e1)p), (ψα)∗((e2)p) of TqS (that is, the basis ∂u |p, ∂v |p of 0 TqS) to be positively oriented. Then a basis B of TqS is positively oriented for the orientation determined by the orienting atlas {ψα : Uα → S}α∈A if ∂ψα ∂ψα and only if det TB0,B > 0, where B = ∂u |p, ∂v |p. ∂ψα ∂ψα Thus we have a choice of a positively oriented basis ∂u |p, ∂v |p of TqS which varies continuously with the point q ∈ S. In practice one rarely uses the above definition of an orientation but rather works with computationally easier to handle but equivalent descriptions or orientability and orientation.

Deﬁnition 2.2 (Outward unit normal). Let {ψα : Uα → S}α∈A be an 3 orienting atlas on a surface S ⊆ R . For q = ψα(p) (where α ∈ A and p ∈ Uα) put

∂ψα ∂ψα ∂u |p × ∂v |p Nq := ∂ψα ∂ψα || ∂u |p × ∂v |p||

Then ||Nq|| = 1, Nq ⊥ TqS and for a basis B = b1, b2 of TqS the basis B is positively oriented for the orientation determined by the orienting atlas 3 {ψα : Uα → S}α∈A if and only if the basis b1, b2,Nq of R is positively 3 oriented with respect to the standard orientation on R . The vector Nq is said to be the outward unit normal of S at q with respect to the orientation determined by the orienting atlas {ψα : Uα → S}α∈A. 3 Theorem 2.3. Let S ⊆ R be a surface. Then the following conditions are equivalent: (1) S is orientable. 3 (2) There exists a continuous vector field W : S → R such that for every q ∈ S we have ||W (q)|| = 1 and W (q) ⊥ TqS. 3 (3) There exists a continuous vector field W : S → R such that for every q ∈ S we have ||W (q)||= 6 0 and W (q) ⊥ TqS. (4) There exists a continuous (with respect to q ∈ S) choice of an orientation on TqS, that is, there exist continuous vector fields V1,V2 : 3 S → R such that for every q ∈ S The pair V1(q),V2(q) is a basis of TqS. (5) The surface S does not contain a copy of the Möbiusband. 4

As a practical matter, we usually specify the orientation on S by providing a continuous unit normal vector field W on S as in part (2) (or a non vanishing continuous normal vector field as in part (3)) of the above theorem. Then a basis b1, b2 of TqS is positively oriented for the orientation on S determined by W if and only if b1, b2,W (q) is a positively oriented basis of 3 3 R with respect to the standard orientation on R . The following key fact provides a rich source of examples of orientable surfaces arising from the Implicit Function Theorem: 3 Theorem 2.4. Let f : R → R be a smooth function, let c ∈ R and let 3 S = {(x, y, z) ∈ R |f(x, y, z) = c}. Suppose that S 6= ∅ and that for every ∂f ∂f ∂f q ∈ S we have grad(f)|q = ( ∂x |q, ∂y |q, ∂z |q) 6= (0, 0, 0). Then: 3 (1) S is an orientable surface in R (2) For every q ∈ S we have grad(f)|q ⊥ TqS. Thus in the above situation we can use grad(f) as a non vanishing normal vector field to define an orientation on S, for which Nq = grad(f)|q/||grad(f)|q|| (where q ∈ S) is the outward unit normal. For this orientation a basis B = b1, b2 of TqS is positively oriented if and only if b1, b2, grad(f)|q is is 3 positively oriented with respect to the standard orientation on R , that is, if and only if (b1 × b2) · grad(f)|q > 0.