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Extrinsic Differential Geometry

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Extrinsic Differential Geometry Extrinsic Differential Geometry J.W.R Most Recent Revision: April 18, 2008 2 Euclidean Space This is the arena of Euclidean geometry; i.e. every figure which is studied in Euclidean geometry is a subset of Euclidean space. To define it one could proceed axiomatically as Euclid did; one would then verify that the axioms characterized Euclidean space by constructing “Cartesian Co-ordinate Systems” which identify the n-dimensional Euclidean space En with the n-dimensional numerical space Rn. This program was carried out rigorously by Hilbert. We shall adopt the mathematically simpler but philosophically less satisfying course of taking the characterization as the definition. We shall use three closely related spaces: n-dimensional Euclidean space En, n-dimensional Euclidean vector space En; and the space Rn of all n-tuples of real numbers. The distinction among them is a bit pedantic, especially if one views as the purpose of geometry the interpretation of calculations on Rn. We can take as our model of En any n-dimensional affine subspace of some numerical space Rk (k > n); the vector space En is then the unique fector subspace of Rk for which: En = p + En for p ∈ En. (Note that En contains the “preferred” point 0 while En has no preferred point; each p ∈ En determines a different bijection v → p + v from En n n onto E .) Any choice of an origin p0 ∈ E and an orthonormal basis e1, . , en for En gives a bijection: n n 1 n X i R → E :(x , . , x ) 7→ p0 + x ei i (the inverse of which is) called a Cartesian co-ordinate system on En. Such space En and En would arise in linear algebra by taking En to be the space of solutions of k − n independent inhomogeneous linear equations in k unknowns while En is the space of solutions of the corresponding homogeneous equations. We now give precise definitions. Definition 1. The orthogonal group O(n) of Rn is the group of all n × n matrices a whose transpose is their inverse: a ∈ O(n) ⇐⇒ aa∗ = e where e is the identity matrix. An equivalent characterization is that: hax, ayi = hx, yi for all x, y ∈ Rn.A group R(n) of rigid motions of Rn is the group generated by O(n) and the group of translation, thus for a˜ : Rn → Rn we have: a˜ ∈ R(n) ⇐⇒ a˜(x) = ax + v (∀x ∈ Rn) for some a ∈ O(n), v ∈ Rn. 3 x ¢ e • ¢ 2 ¢ ¢ ¢ • •¢ 0 e1 Exercise 2. Show thata ˜ : Rn → Rn is a rigid motion, i.e. an element of R(n), if and only if it is an isometry: ka˜(x) − a˜(y)k = kx − yk n for all x, y ∈ R . Hint: You will need the following key lemma: Let ei (i = 1, . , n) be the point given by ei = (0,..., 0, 1, 0,..., 0) with 1 in the i-th slot. Assume x, y ∈ Rn satisfy the n + 1 equations: kx − 0k = ky − 0k, kx − eik = ky − eik i = 1, . , n. Then x = y. Let En be a set. Call two bijections x : En → Rn and y : En → Rn equivalent iff the transformation yx−1 : Rn → Rn is a rigid motion, i.e. yx−1 ∈ R(n). Definition 3. An (n-dimensional) Euclidean space is a set En together with an equivalence class of bijections x : En → Rn; the elements of the equiva- lence class are called Cartesian Co-ordinate systems of the Euclidean space. (Note that by definition any Euclidean space is a manifold diffeomorphic to Rn.) n Definition 4. Call two pairs of points (p1, q1), (p2, q2) from E equivalent iff for some (and hence every) cartesian co-ordinate system we have: x(q1) − x(p1) = x(q2) = x(p2) . Denote the equivalence class of the pair (p, q) by q − p and call it the vector from p to q. Denote the set of all such vectors by En and call it the Euclidean vector space associated to En. The cartesian co-ordinate system x : En → Rn n n induces a map x∗ : E → R by: x∗(q − p) = x(q) − x(p) . Define the operations of vector addition, scalar multiplication, and inner product n in E by declaring that x∗ intertwine these operations with the corresponding ones on Rn; the operations are well-defined (i.e. independent of x) since yx−1 = −1 a˜ ∈ R(n) implies that y∗x∗ = a ∈ O(n). 4 Definition 5. If we fix p ∈ En we obtain a bijection: En → En : q → q − p ; denote the inverse bijection by: En → En : v 7→ p + v . The set of all transformations: En → En : p → p + v form a group called the translation group of En; it is naturally isomorphic to the additive group of the vector space En and will be denoted by the same notation. Definition 6. The groups n −1 n −1 R(E ) = x R(n)x, O(E ) = x∗ O(n)x∗ are independent of the choice of cartesian co-ordinates x : En → Rn. The group R(En) is also set of all transformations a˜ : En → En which preserve the distance function (p, q) 7→ kp − qk = php − q, p − qi; it is called the group of rigid motions of En. The group O(En) is the groups n n n of orthogonal linear transformations of E . For p ∈ E let R(E )p denote the isotropy group: n n R(E )p = {a˜ ∈ R(E ) :a ˜(p) =} . Exercise 7. The translation subgroup En is a normal subgroup on R(En): a˜Ena˜−1 = En fora ˜ ∈ R(En). For each fixed p ∈ En the bijection: En → En : v 7→ p + v n n n intertwines the group O(E ) acting on E with the group R(E )p. The sub- n n n group R(E )p is not normal in R(E ) but every element of R(E ) is uniquely n n expressible as the product of an element in R(E )p and an element of E : n n n n n R(E ) = E · R(E )p, E ∩ R(E )p = {identity}. Remark 8. One summarizes this exercise by saying that R(En) is the semi-direct product of O(En) and En and writing: R(En) =∼ O(En) ./ En . 5 Choosing co-ordinates gives the analogous assertion: R(n) =∼ O(n) ./ Rn which is nothing more than a fancy way of saying thata ˜ ∈ R(n) has form a˜(x) = ax + v for some a ∈ O(n) and v ∈ Rn. There is an important point to be made here. The embeddings of O(En) into R(En) depends on the choice of the “origin” p while the embedding of En into R(En) is independent of any such choice. The subgroup En of R(En) has an “invariant interpretation” and this accounts for why it is normal. (It also explains why normal subgroups are sometimes called “invariant subgroups”; their definitions are independent of the choice of co-ordinates.) Definition 9. An m-dimensional affine subspace of En is a space Em of form Em = p + Em where Em is an m-dimensional vector subspace of En. It is again a Euclidean space in the obvious way: an (inverse of) a cartesian co-ordinate system is a map: m m 1 m X i R → E :(x , . , c ) 7→ p + x ei i m m where p is any point of E and e1, . , em is any orthonormal basis for E . Notation Let M = M m be an m-dimensional submanifold of n-dimensional Euclidean space En. Denote the space of vectors of En by En. Identify the tangent space ⊥ n TpM and the normal space TP M to M at p with vector Subspaces of E : n TpM = {γ˙ (0)|γ : R → M, γ(0) = p} ⊆ E ⊥ n Tp M = {u ∈ E |hu, TpMi = 0} . Denote by X (M) and X ⊥(M) the space of (tangent) vector fields and normal (vector) fields on M respectively. Thus for X, U : M → En we have: X ∈ X (M) ⇐⇒ X(p) ∈ TpM (∀p ∈ M) ⊥ ⊥ U ∈ X (M) ⇐⇒ U(p) ∈ Tp M (∀p ∈ M) More generally for any map ϕ : N → M of a manifold N into M we denote by X (ϕ) (resp. X ⊥(ϕ)) the space of all vector fields (resp. normal fields) along ϕ. Thus for: X, U : N → En we have: X ∈ X (ϕ) ⇐⇒ X(q) ∈ Tϕ(q)M (∀q ∈ N) 6 and: ⊥ ⊥ U ∈ X (ϕ) ⇐⇒ U(q) ∈ Tϕ(q)M (∀q ∈ N) . Such fields are especially important when ϕ is a curve (i.e. N is an open interval in R). Note that X (M) = X (ϕ) and X ⊥(M) = S⊥(ϕ) when ϕ : M → M is the identity. Further note that X ◦ ϕ ∈ X (ϕ) when X ∈ X (M) and ϕ : N → M. n We denote by T˜pM ⊂ E the affine tangent space at p ∈ M: T˜pM = p + TpM. Note the natural isomorphism: TpM → T˜pM : v 7→ p + v . We introduce T˜pM because it is more natural to draw tangent vectors to M at p with their tails at p rather than translated to some artificial origin. The first fundamental form is the field which assigns to each p ∈ M the bilinear map: 2 gp ∈ L (TpM; R) given by: gp(v, w) = hv, wi n for v, w ∈ TpM ⊂ E . Second Fundamental Form For each p ∈ M denote by Π(p) ∈ L(En) the orthogonal projection of En on TpM. It is characterized by the three equations: 2 n ∗ Π(p) = Π(p), Π(p)E = TpM, Π(p) = Π(p).
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