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1. Klein 1.1. affine , ; define Euclidean motions as dis- tance preserving set maps. Proposition. (1) The Euclidean motions are exactly the maps of the form f(x) = Ax + b for A ∈ O(n) and b ∈ Rn. (2) Any Euclidean motion f is a diffeomorphism and writing f(x) = Ax + b as in (1), we have Df(x) = A for all x ∈ Rn. In particular, f is an isometry for the standard Riemannian metric on Rn. Proof. (1) Evidently, x 7→ Ax + b is a Euclidean motion. Conversely, suppose that f : Rn → Rn is a Euclidean motion and define A : Rn → Rn by A(x) := f(x) − f(0). Putting b := f(0), we then have f(x) = A(x) + b, so we have to show that A is an orthogonal linear map. By definition, we obtain kA(x) − A(y)k = kf(x) − f(y)k = kx − yk, and since A(0) = 0, this also shows that kA(x)k = kxk. The polar- 1 2 2 2 isation formula hx, yi = 2 (kxk + kyk − kx − yk ) then shows that hA(x),A(y)i = hx, yi for all x, y ∈ Rn. In particular, for an orthonor- n mal {e1, . . . , en} of R also {A(e1),...,A(en)} is an orthonormal n P basis. Expanding x ∈ R in the basis {ei}, we obtain x = ihx, eiiei. On the other hand, expanding A(x) in the {A(ei)}, we obtain P P A(x) = ihA(x),A(ei)iA(ei) = ihx, eiiA(ei). P P Together with the above, this shows that A( xiei) = xiA(ei), so A is linear, and since we have already observed that A is compatible with the inner product, this shows that A ∈ O(n). (2) is easy.  ∼ n n Euclidean motions form a group Euc(n) = R × O(n), can view R as Euc(n)/O(n). 1.2. Affine motions, and their to affine lines, affine space An as Aff(n)/GL(n). Realization of An as affine in Rn+1 and resulting realization of Aff(n) as a of GL(n + 1). Analogous realization of Euc(n). Structure of euc(n) and Lie bracket, euc(n) ∼= Rn ⊕ o(n) as a representation of O(n) ⊂ Euc(n). Completion of Rn to RP n, viewing RP n as GL(n + 1, R)/P and projective . 1.3. Recall properties of Maurer–Cartan form. Let ω ∈ Ω1(G, g) be the Maurer–Cartan form on a Lie group G. For a closed subgroup H ⊂ G consider the homogeneous space G/H and let p : G → G/H be the canonical . For each g ∈ G the map ω(g): TgG → g induces a linear ϕg : TgH (G/H) → g/h which is characterized by ϕg(Tgp·ξ) = ω(ξ) + h. For h ∈ H, we have −1 ϕgh = Ad(h ) ◦ ϕg. 1 2

Let us specialize to G = Euc(n) and H = O(n). We have seen that euc(n) = Rn ⊕ o(n) and this decomposition is under Ad(h) for each h ∈ H. According to this splitting, we can write ω = θ ⊕ γ for θ ∈ Ω1(G, Rn) and γ ∈ Ω1(G, o(n)). Since the decomposition g = Rn ⊕ o(n) is H–invariant, the two components θ and γ are in- dividually H–equivariant. Explicitly, this means for h ∈ H = O(n) that (rh)∗θ = h−1 ◦ θ and ((rh)∗γ)(ξ) = hγ(ξ)h−1. In particular, we n obtain an identification of g/h with R . Viewing ϕg as a linear isomor- n phism TgH (G/H) → R , it is characterized by ϕg(Tgp·ξ) = θ(ξ) and −1 we get ϕgh = h ◦ ϕg. We can pull back the standard inner product on Rn to the space TgH (G/H) via ϕg. Since each h ∈ H is orthogonal, the result is independent of the choice of g ∈ gH. For a local smooth section σ of p and a vector field ξ ∈ X(G/H) we obtain T p ◦ T σ ◦ ξ = ξ and hence ∗ ϕσ(x)(ξ(x)) = θ(σ(x))(Txσ·ξ(x)) = σ θ(ξ)(x). Hence the inner product of ξ, η ∈ X(G/H) can be locally written as hσ∗θ(ξ), σ∗θ(η)i, which evidently is a smooth function. Hence we have obtained a Riemannian metric on G/H. For each g ∈ G, we have the map `g : G/H → G/H which is characterized by `g ◦ p = p ◦ λg. Left invariance of the Maurer–Cartan ∗ 0 form implies that (λg) θ = θ for all g ∈ G. Now for g, g ∈ G and ξ ∈ Tg0 G we obtain T `g ·T p·ξ = T p·T λg ·ξ and using this, we compute ∗ 0 ϕgg0 (T `g ·T p·ξ) = ϕgg0 (T p·T λg ·ξ) = ((λg) θ(g ))(ξ) 0 = θ(g )(ξ) = ϕg0 (Tp ·ξ).

This shows that the map T `g : Tg0H (G/H) → Tgg0H (G/H) is orthogo- nal, i.e. that G acts on G/H by isometries. In particular, our Riemann- ian metric is invariant under translations, and hence coincides with the standard Riemannian metric on Rn. Proposition. Let U, V ⊂ Rn be connected open and let f : U → V be an isometry for the Riemannian metrics induced from Rn. Then f is the restriction of a uniquely determined Euclidean motion of Rn. Proof. Put U˜ := p−1(U) ⊂ G. We first claim that there is a unique smooth function F : U˜ → G such that • F (gh) = F (g)h for all g ∈ G and h ∈ H • p ◦ F = f ◦ p • F ∗θ = θ

Take a x ∈ G/H. For g ∈ G with x = gH we have the map ϕg : n Tx(G/H) → R , which by construction is an orthogonal isomorphism. −1 Since ϕgh = h ◦ ϕg, the set of all these maps is exactly the set of all orthogonal between the two spaces. Now suppose that −1 −1 n x ∈ U and fix g ∈ p (x). Then ϕg ◦ (Txf) : Tf(x)(G/H) → R 3 is an orthogonal isomorphisms, so there is a unique element F (g) ∈ −1 −1 p (f(x)) such that ϕg ◦ (Txf) = ϕF (g). Equivalently, we can write the defining property of F (g) as ϕg = ϕF (g) ◦ Txf. In this way, we obtain a well defined set map F : U˜ → G, which by construction satisfies p ◦ F = f ◦ p. For h ∈ H we then get

−1 −1 ϕF (gh) ◦ Txf = ϕgh = h ◦ ϕg = h ◦ ϕF (g) ◦ Txf = ϕF (g)h ◦ Txf, which shows that F (gh) = F (g)h. To see that F is smooth, consider local smooth sections σ andσ ˆ of p : G → G/H which are defined locally around x respectively f(x). Then there is an open neighborhood W of x in G/H such that both σ andσ ˆ ◦ f are defined on W . Then both σ∗θ and (ˆσ ◦ f)∗θ are smooth Rn valued one forms on W , whose values are orthogonal. Hence we can define a smooth map ψ : W → O(n) by

ψ(y)(v) := ((ˆσ ◦ f)∗θ)(y)((σ∗θ(y))−1(v)).

This implies that for ξ ∈ Ty(G/H) we obtain

∗ −1 ∗ ∗ ϕσ(y) = σ θ(y) = (ψ(y)) ◦ (f (ˆσ θ))(y) −1 ∗ −1 = (ψ(y)) ◦ (ˆσ θ)(f(y)) ◦ Tyf = (ψ(y)) ◦ ϕσˆ(f(y)) ◦ Tyf

= ϕσˆ(f(y))ψ(y) ◦ Tyf.

Hence we conclude that F (σ(y)) =σ ˆ(f(y))ψ(y) and thus F (σ(y)h) = σˆ(f(y))ψ(y)h for all y ∈ W . Since the map W × H → p−1(W ) defined by (y, h) 7→ σ(y)h is a diffeomorphism, this shows that F is smooth. For g ∈ G and ξ ∈ TgG we now compute

θ(F (g))(TF ·ξ) = ϕF (g)(T p·TF ·ξ) = ϕF (g)(T f ·T p·ξ)

= ϕg(T p·ξ) = θ(g)(ξ), which shows that F ∗θ = θ and completes the proof of the claim. Now we claim that we even have F ∗ω = ω for the Maurer–Cartan form ω. Since the inclusion i : U˜ → G also satisfies i∗ω = ω and U is connected, there is an element g0 ∈ G such that F = λg0 ◦ i which ∗ implies that f = `g0 |U and completes the proof. Since F θ = θ, we get F ∗ω − ω = F ∗γ − γ, and this has values in h ⊂ g. Moreover, consider A ∈ h and the corresponding left invariant vector field LA ∈ X(G). d By definition, exp(tA) ∈ H and LA(g) = dt |t=0g exp(tA). But then F (g exp(tA)) = F (g) exp(tA) and differentiating at t = 0 show that ∗ TgF ·LA(g) = LA(F (g)). This implies that (F ω − ω)(g) vanishes on the subspace {LA(g): A ∈ h}. Consequently, there is a linear map m Φg : R → h such that

∗ (F γ − γ)(g)(ξ) = Φg(θ(ξ)) 4 for all ξ ∈ TgG. Now that last ingredient to use is that ω satisfies the Maurer–Cartan equation. Vanishing of the Rn–component reads as 0 = dθ(ξ, η) + [γ(ξ), θ(η)] + [θ(ξ), γ(η)] = dθ(ξ, η) + γ(ξ)(θ(η)) − γ(η)(θ(ξ)). Of course, also F ∗ω satisfies the Maurer–Cartan equation, and since F ∗θ = θ we get an analogous equation with γ replaced by F ∗(γ). Subtracting these two, we see that 0 = (F ∗γ − γ)(ξ)(θ(η)) − (F ∗γ − γ)(η)(θ(ξ))

= Φg(θ(ξ))(θ(η)) − Φg(θ(η))(θ(ξ)). Any element in Rn can be written as θ(ξ) for an appropriate tangent n vector ξ, so we conclude that the map Φ = Φg : R → o(n) satisfies Φ(a)(b) = Φ(b)(a) for all a, b ∈ Rn. Taking another point c ∈ Rn, we now compute hΦ(a)(b), ci = hΦ(b)(a), ci = −ha, Φ(b)(c)i = −ha, Φ(c)(b)i = hΦ(c)(a), bi = hΦ(a)(c), bi = −hΦ(a)(b), ci This shows that Φ = 0, which completes the proof.  1.4. Klein geometries. Definition of (G, H). Basic properties: connectedness of G/H, effectivity, infinitesimal effectivity, characterization of as max. subgroup of G contained in H. Interpretation in terms of base map. Examples: Determine the H–module g/h in each case. Sn = O(n + 1)/O(n); Hn = O(n, 1)/O(n); S2n+1 as a homoge- neous space of U(n + 1) or SU(n + 1); RP n and Sn as homogeneous spaces of GL(n + 1, R); Sn as a homogeneous space of SO(n + 1, 1); GL(n, R)/O(n); Grassmanians as homogeneous spaces of GL(n, R) and of O(n).

1.5. Canonical curves. via exponential map. Reductive Klein ge- ometries and geodesics. Remark on normal coordinates.

2. Bundles 2.1. Fiber bundles. Fiber bundle charts, fiber bundles; (local) sec- tions, defintion of Γ(E) existence of local smooth sections, bundle pro- jections are surjective submersions, fibers as submanifolds; morphisms and isomorphisms of fiber bundles, trivial bundles, local triviality. Examples Products, tangent bundles, p : G → G/H. Lemma. Let E be a set, M and S smooth and p : E → M a set map. Suppose that there is an open covering {Uα : α ∈ I} of −1 M together with bijective maps ϕα : p (Uα) → Uα × S such that pr1 ◦ϕα = p|Uα . Suppose further that for each α, β ∈ I such that Uαβ := 5

−1 Uα ∩ Uβ 6= ∅ the map ϕα ◦ ϕβ : Uαβ × S → Uαβ × S is given by (x, y) 7→ (x, ϕαβ(x, y)) for a smooth function ϕαβ : Uαβ × S → S. Then E can be uniquely made into a smooth in such a way that {(Uα, ϕα)} is a fiber bundle . sketch of proof. By passing to intersection with charts and then to a countable subcover, we may assume that {Uα} is a countable set of charts for M. Let uα be the corresponding mappings, so {Uα, uα} is a countable atlas for M. Endow E with the initial topology with respect to the maps ϕα. It is easy to see that this topology is Hausdorff. (Points in different fibers can be separated by preimages of open subsets of M, and points in one fiber can be separated by open subsets in S.) Further, since M and S are second countable, it follows that E is scond countable. −1 Fix a countable atlas {Vβ, vβ} for S and use the sets ϕα (Uα × Vβ) and the maps (uα × vβ) ◦ ϕα as charts for E. Using the assumption −1 on the maps ϕα ◦ ϕβ , one easily shows that the chart changes are smooth. 

2.2. Bundles with structure group. Fix a left action of a Lie group G on S. Defintion of G–compatible charts, G–atlasses, equivalence of G–atlasses and fiber bundles with structure group G. Examples TM as a bundle with structure group GL(n, R). G → G/H as a bundle with structure group H.

2.3. Vector bundles. Fiber bundle with fiber a V and structure group GL(V ). Each fiber is a vector space, Γ(E) is vector space and module over C∞(M, R), existence of global smooth sections. homomorphisms. Complex vector bundles and homo- morphisms. Action of vector bundle hommomorphisms (covering the identity) on smooth sections. Proposition. Let E → M and F → M be vector bundles. Then a map Φ : Γ(E) → Γ(F ) comes from a vector bundle homomorphism if and only if it is linear over C∞(M, R). Examples T f : TM → TN, distributions, the canonical bundles on RP n and on Grassmannians.

2.4. Principal fiber bundles. definition, fibers are not groups, pric- ipal right action. Morphisms of principal bundles, reductions of structure group, gauge transformations. Examples G → G/H, the set of all bases of a vector space, the frame bundle of a vector bundle. 6

Lemma. (1) If F : E → E˜ is a morphism of principal bundles and u ∈ E is a point, then F (u) determines the values of F on the fiber containing u. (2) Any gauge transformation is an isomorphism.

Example: The frame bundle of a manifold. Reductions of structure group to O(n) are equivalent to Riemannian metrics.

2.5. Cocycles of transition functions. The cocyle defined by a principal bundle. Isomorphic bundles give rise to cohomologous cocy- cles. Construction of a principal bundle with given cocycle of transition functions.

2.6. G–structures. Defintion

Theorem. Let i : G → GL(n, R) be a homomorphism such that i0 : g → gl(n, R) is injective, let M be a smooth manifold of n and let p : E → M be a principal G–bundle. Then reductions of structure group Φ: E → PM are in bijective correspondence with one–forms θ ∈ Ω1(E, Rn) such that • θ is strictly horizontal, i.e. ker(θ(u)) = ker(Tup) for all u ∈ E • θ is G–equivariant, i.e. ((rg)∗θ)(u) = g−1 ◦ θ(u) for all g ∈ G and u ∈ E. Morphisms of G–structures, interpretation for Riemannian metrics. Lemma. A morphism of G–structres is determined by its base map up to a function into the discrete group ker(i). Interpretation of the first part of Proposition 1.3

2.7. Pullbacks. Pullbacks, definition and universal property; pull- back of sections; preserve the subclasses of vector bundles and of prin- cipal bundles. Proposition. Let E → M and E˜ → M˜ be vector bundles and let F : E → E˜ be a vector bundle homomorphism over f : M → M˜ . If for ˜ each x ∈ M, the restriction Fx : Ex → Ef(x) is a linear isomorphism, then F induces an isomorphism E → f ∗E˜.

Application Tangent bundle of a submanifold of Rn (or of a Lie group G) as a pullback of the tautological bundle over a Grassmannian. More generally for arbitrary bundles over compact manifolds. Remarks on topological aspects of bundle theory.

2.8. Fibered products. Definition and universal property of fibered products. The inverse of the principal right action. A principal bundle which admits a global section is trivial. 7

2.9. Associated bundles. Definition of associated bundles. Proposition. Let p : E → M be a G–principal bundle, G × S → S a smooth left action on a manifold S, E ×G S = E[S] the associated bundle and q : E × S → E ×G S the obvious projection. (1) π : E ×G S → M is a smooth fiber bundle with typical fiber S and structure group G, and the map q is a surjective submersion. (2) There is a smooth map τS : E ×M E[S] → S which is uniquely characterized by the property that for z ∈ E[S] and u ∈ E with π(z) = −1 p(u) we have z = q(u, τS(u, z)). In particular, τS(u·g, z) = g ·τS(u, z). (3) q : E × S → S ×G S is a G–principal bundle. Let p : E → M be a G–principal bundle and G × S → S a smooth left action. Then a smooth map f : E → S is said to be G–euqivariant if and only if f(u·g) = g−1 ·f(u) for all u ∈ E and g ∈ G. The space of all such maps is denoted by C∞(E,S)G.

Corollary. The space Γ(E ×G S) of smooth sections of an associated bundle is naturally isomorphic to C∞(E,S)G. 2.10. Generalized Frame bundles. Proposition. Let π : E → M be a fiber bundle with fiber S and structure group G which acts effectively on S. Then there is a unique (up to isomorphism) G–principal bundle p : F → M such that E ∼= F ×G S as a bundle with structure group G. 2.11. Constructions with vector bundles. Let π : E → M be a vector bundle with fiber V and frame bundle p : P(E) → M. Then P(E) ×GL(V ) V is naturally isomorphic to E. The isomorphism is in- duced by (ϕ, v) 7→ ϕ(v). Constructions via associated bundles: E ⊕ F , L(E,F ), E∗, E ⊗ F , SkE,ΛkE, etc. Remark on K–theory. Kernels and images of (appropriate) vector bundle homomorphisms of constant rank are smooth subbunldes. Subbundles and quotients, exact sequences of vector bunldes. 2.12. Functorial properties of associated bundles. Functoriality in both arguments.

3. Homogeneous bundles and invariant sections 3.1. Homogeneous bundles. Definition of homogeneous fiber bun- dles, vector bundles, and principal bundles. Basic Examples: Natural bundles, G → G/H, bundles associated to a homogeneous principal bundle. Theorem. (1) Let π : E → G/H be a homogeneous fiber bundle (re- spectively vector bundle), o = eH ∈ G/H the base point and Eo = 8

π−1(o) the fiber over o. Then the G–action on E naturally restricts to an action (respectively a representation) of H on Eo and E is naturally isomorphic to G ×H Eo as a homogeneous bundle. (2) Passing to associated bunldes induces equivalences of categories be- tween left H–spaces and homogeneous fiber bundles as well as between representations of H and homogeneous vector bundles. (3) Any homogeneous principal K–bundle over G/H is of the form G ×H K, where H acts on K via h·k = i(h)k for a homomorphism ∼ i : H → K. Denoting this bundle by G ×i K, we have G ×i K = G טi K if and only if i and ˜i are conjugate. 3.2. Sections of homogeneous bundles. The natural action of G on sections of a homogeneous bundle. Induced representations both in the picture of sections and of equivariant functions. Theorem. Consider the homogeneous bundle π : E → G/H corre- sponding to a given left action H × S → S. Then σ 7→ σ(o) induces a between G–invariant elements in Γ(E) and H–invariant ele- ments in S. More generally, let evo : Γ(E) → Eo denote the evaluation at o. Then given any left action G × X → X on a smooth manifold X, mapping F to evo ◦ F induces a bijection between the set of G–equivariant maps X → Γ(E) such that evo ◦ F : X → Eo is smooth and the set of smooth H–equivariant maps X → Eo. Corollary (Frobenius reciprocity). Let V be a finite dimensional rep- G resentation of H and IndH (V ) the induced representation of G on G Γ(G×H V ). Further, let W be any representation of G and let ResH (W ) be the restriction of W to H. Then G ∼ G HomG(W, IndH (V )) = HomH (ResH (W ),V ) 3.3. Examples. Invariant Riemann metrics: For a homogeneous space G/H, we ask whether there is a Riemannian metric γ on G/H such that G acts on G/H by isometries. A metric γ can be considered as a smooth section of the bundle S2T ∗M. For a vector field ξ ∈ X(G/H) ∗ one easily verifies directly that g·ξ = (`g−1 ) ξ for each g ∈ G. Using ∗ this, one obtains that for any tensor field t on G/H we get g·t = (`g−1 ) t by functoriality. In particular, for a Riemannian metric γ, and vector fields ξ and η, we get ∗ −1 (g·γ)(x)(ξ, η) = ((`g−1 ) γ)(x)(ξ, eta) = γ(g ·x)(T `g−1 ·ξ, T `g−1 ·η) But by definition, this means that g·γ = γ if and only if each of the maps T `g−1 is orthogonal. In particular, G acts by isometries for γ if and only if γ ∈ Γ(S2T ∗(G/H)) is a G–invariant section. Hence we conclude Proposition. G–invariant Riemannian metrics on G/H are in bijec- tive correspondence with inner products on g/h = To(G/H), which are 9 invariant for the H–action induced by Ad. In particular, a G–invariant inner product exists if and only if the subgroup Ad(H) ⊂ GL(g/h) has compact .

In particular, this shows that if H is compact, then there always is an invariant Riemannian metric on G/H. Viewing Sn as SO(n+1)/SO(n), ∼ n we know from 1.4 that g/h = R as an H–module. In particular, up to scale, there is only one SO(n)–invariant inner product on g/h. Hence the SO(n + 1)–invariant Riemannian metric on Sn is unique up to constant rescalings. The same argument shows that the flat metric on Rn is (up to constant scale) the unique Riemannian metrich which is invariant under the Euclidean group, and the hyperbolic metric on Hn = SO(n, 1)/SO(n) is uniquely determined up to scale by invariance under SO(n, 1). On the other hand, we can look at S2n+1 as the homogeneous space ∼ n U(n+1)/U(n). Then we know that g/h = C ⊕R as a representation of U(n). This easily implies that there is a two–parameter family of U(n+ 1) invariant Riemannian metrics on S2n+1 obtained by independently rescaling the standard inner products on the two factors.

Decomposing spaces of functions or sections: This is a short out- look on how one proceeds in understanding induced representations. For simplicity, let us focus on the example of Sn = SO(n + 1)/SO(n). The simplest example of an induced representation then is provided by starting from the trivial representation R of SO(n). By definition, SO(n+1) ∞ n IndSO(n) (R) = C (S , R) with the action of SO(n + 1) given by g·f := f ◦ `g−1 . Frobenius reciprocity tells us that for any represen- ∞ n tation V of SO(n + 1), the space HomG(V,C (S , R)) is isomorphic to HomH (V, R). By compactness of SO(n + 1), we may assume that V is irreducible. Restricting the representation V to SO(n), it is not irreducible any more, but it splits into a direct sum of irreducible rep- n(V ) resentations. The space HomH (V, R) is simply R , where n(V ) is the number of trivial factors in this decomposition. For example, taking V = R, we of course have n(R) = 1 so

∞ n HomG(R,C (S , R)) = R. This corresponds to the fact that the constant functions on Sn are the only G–invariant functions. Next, consider the defining represen- tation V = Rn+1. Restricted to SO(n), this decomposes as Rn ⊕ R, n+1 n+1 ∞ n so n(R ) = 1, and HomG(R ,C (S , R)) = R. The of any nonzero homomorphism in this family (which then is independent of the choice) is an n + 1–dimensional subrepresentation of C∞(Sn, R) isomorphic to Rn+1. Of course, this subrepresentation is spanned by the the components of the inclusion Sn ,→ Rn+1. 10

More generally, for each k ∈ N, the representation SkRn+1 restricted k i n to SO(n) decomposes as ⊕i=0S R . In particular, each of these repre- sentations contains exactly one copy of the trivial representation. Hence for each k, the representation C∞(Sn, R) contains a unique subrepre- sentation isomorphic to SkRn+1. This is spanned by the restrictions to Sn of homogeneous of degree k on Rn+1. More complete information can be obtained using the Peter–Weyl theorem. Since SO(n + 1) is compact, the functions in which lie in a fi- nite dimensional SO(n+1)–invariant subspace are dense in C∞(Sn, R). By complete reducibility, any such function can be written as a finite sum of elements in the image of a G–equivariant map from an appro- priate finite dimensional irreducible representation V to C∞(Sn, R). Hence thes ideas lead to a complete description of a dense subspace of the representation C∞(Sn, R). Similar arguments apply to more general induced representations. Invariant differential forms. Applications: canonical symplectic struc- ture on coadjoint orbits, of homogeneous spaces of compact groups. 4. Connections and curvature 5. Invariant connections