CHAPTER 3 G-structures 3.1. Geometric structures on vector spaces 3.1.1. Linear G-structures Many types of geometric structures on manifolds M are of infinitesimal type: they are collections of structures on the tangent spaces TxM, ”varying smoothly” with respect to x M (think e.g. about Riemannian metrics on M). Hence, the first step is to understand2 (the meaning of) such structures on general finite dimensional vector spaces V . The basic idea is to characterize the geometric structures of interest via ”the frames adapted to the structure”: for many types of geometric structures on vector spaces V , one can make sense of frames adapted to the given structure; for instance, for an inner-product on V , one looks at frames which are orthonormal with respect to the given inner product; or, for an orientation on V , one looks at oriented frames; etc. This process of defining the notion of frames adapted to a geometric structure depends of course on the type of structures one considers. But, for most of them, this process can be divided into several steps which we now describe. Start with a class truct of geometric structures- for instance: S truct inner products, orientations, complex structures, symplectic structures, etc S 2{ } (to be discuss in more detail later on). Hence, for any finite dimensional vector space V one has the set truct(V ) of truct-structures on V and one is interested in pairs S S (V,σ) consisting of a vector space V and an element σ truct(V ). The steps we men- tioned are: 2S S1: One has a notion of isomorphism of such pairs (V,σ). In particular, we can talk about the automorphism group Aut(V,σ) GL(V). ⇢ S2: One has a a standard model, which is Rn with ”a canonical” truct-structure S n σcan truct(R ). 2S It is a model in the sense that any n-dimensional vector space V endowed with σ n 2 truct(V ) is isomorphic to (R ,σcan) (just that the isomorphism is not canonical!). S S3: Of capital importance is the group of automorphisms of the standard model: n G := Aut(R ,σcan) GLn. ⇢ 81 82 M. CRAINIC, DG-2015 S4: While we identify a frame φ =(φ1,...,φn) of V with the associated linear isomorphism φ : Rn V , the condition that φ is adapted to a given structure ! n σ truct(V ) means that φ is an isomorphism between (R ,σcan) and (V,σ). This gives2S rise to the space of adapted frames Fr(V,σ) Fr(V) ⇢ and the main idea is that σ is encoded in Fr(V,σ). To point out the main properties of Fr(V,σ) we will use the right action of GLn on Fr(V) (page 65) and, for two frames φ and φ0 of V , we denote by [φ : φ0] GL 2 n the resulting matrix of coordinate changes- defined by φ = φ [φ : φ ]. With these: 0 · 0 φ Fr(V,σ), A G= φ A Fr(V,σ). • φ,2φ Fr(V,σ)=2 [φ): φ ]· G.2 • 0 2 ) 0 2 Equivalently, Fr(V,σ)isaG-invariant subspace of Fr(V) on which G acts transi- tively. Note that the previous discussion can be carried out without much trouble in all the examples we will be looking at. However, the general discussion is not very precise- as it appeals to our imagination for making sense of some of the ingredients. Of course, it can be made precise by axiomatizing the notion of ”class truct of geo- metric structures on vector spaces” (e.g. using the language of functors).S However, the philosophy that geometric structures are characterized by their adapted frames leads to a di↵erent approach in which the class of geometric structures truct is associated to the group G and a G-structure on V is encoded in the frameS bundle: Definition 3.1. Let G be a subgroup of GLn(R). A linear G-structure on an n- dimensional vector space V is a subset Fr(V ) S⇢ satisfying the axioms: A1: is G-invariant, i.e.: φ ,A G = φ A . A2:S if φ,φ then [φ : φ ] 2GS. 2 ) · 2S 0 2S 0 2 We denote by truct (V ) the space of linear G-structures on V . S G Remark 3.2. The second axiom implies that all the frames φ induces the same element in the quotient, 2S σ Fr(V)/G. S 2 Conversely, starting from σ (which can be any element in this quotient), can be recovered as the set of all framesS φ Fr that represent σ . In other words,S 2 S truct (V ) = Fr(V)/G. S G Remark 3.3 (isomorphisms; the standard model). One can now check that the pre- vious discussion (the steps S1-S4) can be carried out for the abstractly defined notion of linear G-structures- i.e. the tructG of the previous definition. To do that, note that one can talk about: S 1. isomorphisms between two linear G-structures: given i on Vi, i 1, 2 , an isomorphism between the pairs (V , ) is any linear isomorphismS 2{ } i Si A : V V 1 ! 2 3.1. Geometric structures on vector spaces 83 with the property that (φ ,...,φ ) = (A(φ ),...,A(φ )) . 1 n 2S1 ) 1 n 2S2 2. the standard linear G-structure on Rn can n Fr(R ) SG ⇢ consisting of frames which, interpreted as matrices , are elements of G. (We use here the conventions from pages 64-65. Hence the way that we interpret n a frame φ =(φ1,...,φn) of R as a matrix is by interpreting each vector φi as a column vector: 1 φi [φi]= ... 0 n1 φi so that φ is identified with @ A 1 1 φ1 ... φn [φ]=([φ1],...,[φn]) = ... ... ... 0 n n1 φ1 ... φn @n A In this way the right action of GLn on Fr(R ) will be identified with the usual multiplication of matrices.) With these, one can carry out steps S1-S4 for truct and one can show that, S G indeed, for any tructG(V ) (playing the role of σ in the general discussion at the start), the resultingS2S space of adapted frames Fr(V, )isprecisely . S S 3.1.2. Example: Inner products Recall that a linear metric (inner product) on the vector space V is a symmetric bilinear map g : V V R ⇥ ! with the property that g(v, v) 0 for all v V , with equality only for v = 0. There is an obvious notion of frames≥ of V adapted2 to g: the orthonormal ones, i.e. the ones of type φ =(φ1,...,φn)with (3.4) g(φ ,φ )=δ i, j. i j i,j 8 However, it is instructive to find them going through stepst S1-S4 mentioned at the beginning of this chapter. S1: Given two vector spaces endowed with inner products, (V,g), (V 0,g0), it is clear what an isomorphism between them should be: any linear isomorphism A : V V 0 with the property that ! g0(A(u),A(v)) = g(u, v) u, v V. 8 2 One also says that A is an isometry between (V,g) and (V 0,g0). n S2: The standard model is R with the standard inner product gcan: g (u, v)= u, v = ui vi, can h i · Xi 84 M. CRAINIC, DG-2015 or, in the matrix notation: v1 1 n (3.5) gcan(u, v)=(u ...u ) ... = u [v]. · 0vn 1 · @ A (recall that [v]=vT stands for v interpreted as a column matrix- see page 65). We leave it as an exercise to show that, indeed, any vector space endowed with an inner n product, (V,g), is isomorphic to (R ,gcan). S3: The symmetry group of the standard model becomes ˆ ˆ n G = A GLn(R):gcan(A(u), A(v)) = gcan(u, v) for all u, v R GLn(R). { 2 2 }⇢ Here Aˆ denotes the matrix A interpreted as a linear map (see page 64). Hence, while as column matrices one has [Aˆ(u)] = A [v] (see page 65), as a row matrix Aˆ(u)is v AT . We find that · · T G = A GLn(R):A A = I = O(n). { 2 · } S4: By the general procedure, ”the adapted frames” of (V,g) are those frames φ Fr(V ) with the property that the induced linear isomorphism φˆ : Rn V 2 n ! is an isomorphism between (R ,gcan) and (V,g). It is now easy to see that this happens if and only if φ is orthonormal with respect to g. Hence one ends up with the set of orthonormal frames of (V,g), denoted Fr(V ). Sg ⇢ The following Proposition makes precise the fact that (and explains how) the ”inner product g” is encoded by the associated set of frames . Sg Proposition 3.6. Given an n-dimensional vector space V , g ! S g defines a 1-1 correspondence between: 1. inner products on V . 2. linear O(n)-structures on V . Proof. The main point is that g determines g uniquely: using any φ g, and S i 2S decomposing arbitrary elements of V with respect to φ as u = i u φi, it follows that · P g(u, v)= ui vi. · Xi This shows how to reconstruct g out of any linear O(n)-structure ; the axioms for imply the resulting g does not depend on the choice of φ and thatS = . S Sg S 3.1.3.