LECTURE 2: EUCLIDEAN SPACES, AFFINE SPACES, AND HOMOGENOUS SPACES IN GENERAL 1. Euclidean space n If the vector space R is endowed with a positive definite inner product h, i n we say that it is a Euclidean space and denote it E . The inner product n gives a way of measuring distances and angles between points in E , and this is the fundamental property of Euclidean spaces. Question: What are the symmetries of Euclidean space? i.e., what kinds of n n transformations ϕ : E → E preserve the fundamental properties of lengths and angles between vectors? Answer: Translations, rotations, and reflections, collectively known as “rigid motions.” Any such transformation has the form ϕ(x) = Ax + b n where A ∈ O(n) and b ∈ E . The set of such transformations forms a Lie group, called the Euclidean group E(n). This group can be represented as a group of (n + 1) × (n + 1) matrices as follows: let A b E(n) = : A ∈ O(n), b ∈ n . 0 1 E Here A is an n × n matrix, b is an n × 1 column, and 0 represents a 1 × n n row of 0’s. If we represent a vector x ∈ E by the (n+1)-dimensional vector x , then elements of E(n) acts on x by matrix multiplication: 1 A b x Ax + b = . 0 1 1 1 n Question: Given a point x ∈ E , which elements of E(n) leave x fixed? This question is easiest to answer when x = ~0. It is clear that an element A b A 0 fixes ~0 if and only if b = 0. The set of such elements forms a 0 1 0 1 ~ n subgroup H~0, called the isotropy group of 0 ∈ E , which is clearly isomorphic to O(n). n What about other points x ∈ E ? It seems reasonable to expect that there shouldn’t be anything special about ~0 since we can move any point to any other point via a translation. And in fact, the isotropy group Hx of any point n x ∈ E is also isomorphic to O(n). We can define an explicit isomorphism φ : 9 10 JEANNE NIELSEN CLELLAND I x H → H as follows. Let t = , so that t represents the translation ~0 x x 0 1 x tx(y) = y + x. Then define −1 φ(h) = txhtx . φ(h) clearly fixes x, and it is not difficult to see that φ is an isomorphism. −1 Thus Hx = txH~0 tx , and all the isotropy groups Hx are conjugate in E(n) and isomorphic to O(n). Note that the left cosets txH~0 have the form I x A 0 A x t H = = : A ∈ O(n) . x ~0 0 1 0 1 0 1 n Thus the space E is isomorphic to the set of left cosets of the subgroup O(n) ⊂ E(n). In fact, if we give E(n)/O(n) the quotient topology, then we n ∼ have a diffeomorphism E = E(n)/O(n). 2. Moving frames on Euclidean space n Another way to look at all this is in terms of frames on E . We define a frame n n on E to be a set of vectors (x; e1, . , en) where x ∈ E and {e1, . , en} n is an orthonormal basis for the tangent space to E at x. If we regard the vectors e1, . , en as the columns of a matrix A ∈ O(n), we see that the n set of frames on E is isomorphic to the Euclidean group E(n): the vector x represents the translation component, and the matrix A represents the rotation component. Regarded in this way, we can define a projection map n π : E(n) → E by π(x; e1, . , en) = x. The fiber of this map is the set of all orthonormal frames at x and so is isomorphic to O(n). This map describes E(n) as a principal bundle over n n E with fiber O(n), called the frame bundle of E and sometimes denoted n F(E ). n The vector components x, e1, . , en may all be thought of as E -valued n functions on E(n). Thus their exterior derivatives dx, dei should be T E - valued 1-forms on E(n). Since {e1, . , en} is a basis for the tangent space n to E at each point, these exterior derivatives can be expressed as linear combinations of e1, . , en whose coefficients are ordinary scalar-valued 1- forms. It will become clear over the next several lectures that the power of the method of moving frames lies in expressing the derivatives of a frame i j in terms of the frame itself. We define 1-forms ω , ωi , 1 ≤ i, j ≤ n by the LIE GROUPS AND THE METHOD OF THE MOVING FRAME 11 equations n X i (2.1) dx = ei ω i=1 n X j dei = ej ωi . j=1 n n 1 n Since dx : TxE → TxE is the identity map, the 1-forms ω , . , ω form a n basis for the 1-forms on E . They are often called the dual forms of the frame {e1, . , en}. (For instance, if we take ei to be the ith standard basis vector i i εi for i = 1, . , n, then ω = dx .) The dual forms have the property that ωi(v) = 0 for any vector v which is tangent to the fibers of the projection n π : E(n) → E ; we say that forms with this property are semi-basic for the i projection π. The ωj, on the other hand, form a basis for the 1-forms on each fiber of π. They are often called the connection forms of the frame {e1, . , en}. i j Differentiating equations (2.1) shows that the forms ω , ωi satisfy the struc- ture equations n i X i j (2.2) dω = − ωj ∧ ω j=1 n i X i k dωj = − ωk ∧ ωj . k=1 n We have not yet used the fact that we have a Euclidean strcture on E . Since the ei are orthonormal vectors, we have ( 1 i = j hei, eji = δij = . 0 i 6= j i Differentiating these equations shows that the ωj are skew-symmetric; i.e., i j ωj = −ωi . This can all be described in terms of the Lie group E(n). Let e(n) denote the tangent space to E(n) at the identity element; e(n) is called the Lie algebra of E(n), and it is not difficult to show that a b e(n) = : a is a skew-symmetric n × n matrix and b ∈ n . 0 0 E Let g : E(n) → E(n) represent the identity map, and consider the 1-form ω = g−1 dg. ω is an e(n)-valued 1-form, called the Maurer-Cartan form of E(n), and it is left-invariant. This means that if h ∈ E(n) and Lh : E(n) → E(n) denotes ∗ left multiplication by h, then Lhω = ω. As a consequence, ω is completely 12 JEANNE NIELSEN CLELLAND determined by its values at the identity. Moreover, the matrix entries of ω form a basis for the left-invariant scalar 1-forms on E(n). It is straightforward to check that ω satisfies the Maurer-Cartan equation dω = −ω ∧ ω. i i The entries of ω are exactly the scalar 1-forms ω , ωj. For instance, when n = 3, we have 1 3 1 0 ω2 −ω1 ω 1 2 2 −ω2 0 ω3 ω ω = 3 2 3 . ω1 −ω3 0 ω 0 0 0 0 It is not difficult to show that the Maurer-Cartan equation is equivalent to i i the structure equations given above for the ω , ωj. 3. Affine spaces n n Now we consider a different structure on R . If R is endowed with a volume form dV (i.e., a measure of the volume of the parallelepiped spanned by any n vectors v1, . , vn), we say that it is an affine space (or special affine n space) and denote it A . Affine space has less structure than Euclidean space; there is no inner product, and thus no way of measuring distances or angles between vectors. Question: What are the symmetries of affine space? Since there is less structure that must be preserved, we might expect that the symmetry group n n would be larger. Indeed, the transformations ϕ : A → A which preserve the affine structure are those of the form ϕ(x) = Ax + b n where A ∈ SL(n) and b ∈ A . Just as in the Euclidean case, these trans- formations form a Lie group, called the affine group A(n), which can be represented as A b A(n) = : A ∈ SL(n), b ∈ n . 0 1 A Now the isotropy group H~0 is A 0 H = : A ∈ SL(n) ~0 0 1 and the isotropy groups Hx are all conjugate and isomorphic to SL(n). By n the same reasoning as in the Euclidean case, the affine space A is isomorphic to A(n)/SL(n), the set of left cosets of SL(n) in A(n). n n A frame on affine space A is a set of vectors (x; e1, . , en) where x ∈ A n and {e1, . , en} is any basis for the tangent space to A at x which spans LIE GROUPS AND THE METHOD OF THE MOVING FRAME 13 a parallelepiped of volume 1. (Such a frame is called unimodular.) By the same reasoning as in the Euclidean case, the set of frames is isomorphic to n the affine group A(n).