Riemannian Geometry - University Of Sydney

Lie groupsLet Mn (R) denote the space of n n matrices with real entries.Example 17.3. The general linear groupGL(n, R) {A Mn (R) det A 6 0}2is an open subspace of Rn and hence a manifold of dimension n2 . It is easy to verify that the group operations(g, h) 7 gh and g 7 g 1 are smooth.More intrinsically, GL(n, R) is the space of invertible linear transformations Rn Rn .Example 17.4. The positive general linear groupGL (n, R) {A Mn (R) det A 0}is a Lie subgroup (i.e. a subgroup and a submanifold) of the general linear group and has the same dimension n2since it is an open subset.Exercise 17.5. GL (n, R) is the group of linear transformations Rn Rn which preserve orientation.Recall:Definition 17.6. An orientation of an n-dimensional vector space V is a choice of one of the two possible equivalence classes of ordered basis for V under the equivalence relation defined by:(X1 , . . . , Xn ) (Y1 , . . . , Yn )for ordered bases (X1 , . . . , Xn ) and (Y1 , . . . , Yn ) of V if the n n matrix A defined by(Y1 , . . . , Yn ) (X1 , . . . , Xn )Ahas positive determinant.Example 17.7. The set of symmetric matricesSym(n) {A Mn (R) t A A}is not a Lie group under multiplication since it contains the zero matrix, and so is not a group.It is however a manifold since taking the diagonal and upper triangular entries defines a global diffeomorphism toRk , wheren(n 1).k 1 2 ··· n 2Exercise 17.8. A matrix is symmetric if and only iffor all x, y Rn .hAx, yi hx, AyiExample 17.9. The orthogonal group is defined byO(n) {G Mn (R) t AA I}.Note that this forces the determinant of A to be either 1 or 1. In fact the group O(n) consists of two connectedcomponents.Exercise 17.10. Prove that O(n) is the group of linear transformations Rn Rn which preserve the Euclideaninner product in the sense thatA O(n) hAx, Ayi hx, yifor all x, y Rn .Example 17.11. Consider the special orthogonal groupSO(n) {A Mn (R) det A 1 and t AA I}.8

Exercise 17.12. Prove that SO(n) is precisely the group of linear transformations Rn Rn which preserve theEuclidean inner product and orientation.Example 17.11 (Continued). Moreover definingf : GL (n, R) A7 Sym(n)AA I,twe have SO(n) f 1 (0).The map f is smooth, and furthermore it is a submersion.Definition 17.13. A smooth map f : M n m N n between smooth manifolds is a submersion at p M if dfphas (maximal) rank n.The map is a submersion if it is is a submersion at every point in its domain.Equivalently, dfp is surjective.Example 17.11 (Continued). In our example we have2f : GL (n, R) Rn Sym(n) Rn(n 1)/2openand f being a submersion tells us that it satisfies the criteria of the implicit function theorem.RecallTheorem 17.14 (Implicit Function Theorem). SupposeF : W Rm n Rm Rn Rn(x1 , . . . , xm , y1 , . . . , yn )7 is a smooth map, and that for (a, b) W , 1F1 (a, b) F12 (a, b) . .F21 (a, b)F22 (a, b).(F 1 (x, y), . . . , F n (x, y))······.Fn1 (a, b)Fn2 (a, b).F1n (a, b) F2n (a, b) · · ·Fnn (a, b) imof a andis invertible, where Fji F yj . Write c F (a, b). Then there are open neighbourhoods U RnV R of b and a smooth map g : U V so that for (x, y) U V ,F (x, y) c y g(x).Example 17.11 (Continued). The implicit function theorem tells us then that SO(n) f 1 (0) is a manifold ofdimensionn(n 1)2n2 n2 nn(n 1)n2 .222The group operations (multiplication and taking inverses) are smooth and hence SO(n) is a Lie group.Example 17.15. Similarly, applying the same argument tof : GL(n, R) A7 Sym(n)AA I,twe recognise O(n) as f 1 (0) and hence see that O(n) is a Lie group of dimensionhas two connected components, one of which is SO(n).n(n 1).2As stated above, O(n)Example 17.16. The unitary group U (n) GL(n, C) is the subgroup of the complex general linear group consisting of matrices satisfying t ĀA I.The special unitary group SU (n) consists of those unitary matrices which additionally satisfy det A 1.9

Exercise 17.17. Prove that U (n) is the group of linear transformations Cn Cn which preserve the Hermitianinner product(z, w) z1 w1 · · · zn wnin the sense thatA U (n) (Az, Aw) (z, w) for all z, w Cn .The space of Hermitian symmetric matricesHermSym(n) {A Mn (C) t A A}consists of matrices of the form x1 z12 z13 . . z1nz12x2z 23.z13z23.······.z1nz2n.xn 1zn 1,nzn 1,nxn 2where zij C, xi R and hence is a manifold diffeomorphic to Rn(n 1) n Rn .Exercise 17.18. Show that A HermSym if and only if for all z, w Cn ,(Az, w) (z, Aw).Active LearningQuestion 17.19. By an analogous argument to that used for SO(n), prove that U (n) is a real smooth manifold ofdimension n2 .Note that a unitary matrix satisfiesdet A det A 11and hence det A S .We may characterise SU (n) as the pre-image of 1 under the mapdet : U (n) S 1 .Again det is a submersion and so the implicit function theorem tells us that SU (n) is a smooth real manifold ofdimension n2 1.Definition 17.21. a Lie group G acts on a manifold M if there is a mapG M M(g, p) 7 g · psuch thate·p pfor all p Mand(gh) · p g · (h · p)for allg, h G, p M.(these force the action of each element to be a bijection)Definition 17.21 (continued). if for every g G,g: Mp M7 g·pis smooth then we say that G acts smoothly the isotropy subgroup Gp of G at p is the subgroup fixing p10

G acts transitively on M if for every p, q M there exists g G such that q g · pDefinition 17.22. A homogeneous space is a manifold M together with a smooth transitive action by a Lie groupG.Informally, a homogeneous space “looks the same” at every point.Since the action is transitive, the isotropy groups are all conjugateGg·p gGp g 1and for any p M we can identify the points of M with the quotient space G/Gp .Definition 17.23. A Lie subgroup H of a Lie group G is a subset of G such that the natural inclusion is animmersion and a group homomorphism (i.e. H is simultaneously a subgroup and submanifold).Since the the group action on a homogeneous space is in particular continuous, the isotropy subgroups are closedin the topology of G.Theorem 17.24 (Lie-Cartan). A closed subgroup of a Lie group G is a Lie subgroup of G.Corollary 17.25. An isotropy subgroup Gp of a Lie group G is a Lie subgroup of G.Theorem 17.26. If G is a Lie group and H a Lie subgroup then the quotient space G/H has a unique smoothstructure such that the mapG G/H G/H(g, kH) 7 gkHis smooth.We omit the proof; it is given for example in Warner, “Foundations of Differentiable Manifolds and Lie Groups”,pp 120–124.Corollary 17.27. A homogeneous space M acted upon by the Lie group G with isotropy subgroup Gp is diffeomorphic to the quotient manifold G/Gp where the latter is given the unique smooth structure of the previoustheorem.Lecture 18[Riemannian Geometry – Lecture 18]Riemannian Geometry – Lecture 18 Homogeneous spaces Dr. Emma CarberrySeptember 21, 2015Example 18.1. PropositionThe Lie group SO(n 1) acts smoothly on the unit sphere S n . For every p S n the isotropy subgroup is conjugateto SO(n). Hence S n is a homogeneous space and is diffeomorphic to the quotient manifold SO(n 1)/SO(n).Here we are identifying SO(n) with its image under the embeddingSO(n) SO(n 1) 1 0A 7 0 AExample 18.1 (Continued). Proof: The actionSO(n 1) Rn 1 Rn 1(g, p) 7 g · ppreserves the Euclidean inner product and hence gives an actionSO(n 1) S n S n ,which is clearly smooth.11

Example 18.1 (Continued). To see that the action is transitive, note that it suffices to show that for any p S n , wemay find g SO(n 1) such that ge1 p.But given p S n , we may take v2 , . . . , vn 1 S n such that v1 p, v2 , . . . , vn 1 is an orthonormal basis ofRn 1 with the same orientation as the standard basis.Then g SO(n 1) defined byvi gij ej ,satisfies p ge1 .Example 18.1 (Continued). The isotropy subgroup SO(n 1)e1 (i.e. the subgroup fixing e1 ) clearly contains 1SO(n) .SO(n)Conversely, suppose e1 g · e1 g1j ej . Theng11 1 and g1j 0 for j 6 1.Since g SO(n 1) O(n 1), g t g I, soX(gj1 )2 1jand as g11 1 this gives gj1 0 for j 6 1.Proposition 18.2. Let H be a closed subgroup of the Lie group G and suppose that H and G/H are connected.Then G is connected.Proof: SupposeG U V for 6 U, V open subsets of G.Writing π : G G/H for the projection thenG/H π(U ) π(V ) and 6 π(U ), π(V ) open subsets of G/H.Since G/H is connected, for some g G we havegH π(U ) π(V ).Since G U V thengH (gH U ) (gH V ).Since gH H inherits its topology from G, these are open subsets of gH and since H is connected,(gH U ) (gH V ) 6 soU V 6 so G is connected.Corollary 18.3. SO(n) is connected for all n 1.Proof. We use induction on n.SO(1) is a single point so clearly connected. Then from the above proposition and the fact that S n is connected,the result follows.Corollary 18.4. O(n) has two connected components for all n 1.12

Proof. The mapR : SO(n) O(n)g 7 Agwhere 1 A 1 1·1is a homeomorphism from SO(n) to the subset of O(n) consisting of matrices of determinant 1.O(n) SO(n) R(SO(n))then expresses O(n) as the union of two open, nonempty, disjoint connected subsets.Example 18.5. PropositionThe Lie group O(n 1) acts smoothly on the unit sphere S n . For every p S n the isotropy subgroup is conjugateto O(n). Hence S n is a homogeneous space and is diffeomorphic to the quotient manifold O(n 1)/O(n).Proof: ExerciseActive LearningQuestion 18.6. Show that the Lie group U (n 1) acts transitively on the sphere S 2n 1 , and for each p S 2n 1the isotropy subgroup is conjugate to U (n). Hence the sphere S 2n 1 is a homogeneous space and is diffeomorphicto U (n 1)/U (n).Again, we may replace the unitary group by the special unitary group in this example and hence obtain that S 2n 1is diffeomorphic to SU (n 1)/SU (n).Taking n 1, note that SU (1) consists of a single point, the identity matrix, and hence S 3 is diffeomorphic to theLie group SU (2).In fact S 3 and S 1 are the only spheres which admit Lie group structures.Example 18.8 ( Real projective space, RPn ). Recall that RPn is the space of lines through the origin in Rn 1 .Equivalently,RPn Rn 1 \ {0}, where(X 0 , X 1 , . . . , X n ) λ(X 0 , X 1 , . . . , X n ), λ R \ {0}.We write the equivalence class of (X 0 , X 1 , . . . , X n ) as [X 0 : X 1 : . . . : X n ].Example 18.8 (Continued). The natural projectionRn 1 \ {0} RPn(X 0 , X 1 , . . . , X n ) 7 [X 0 : X 1 : . . . : X n ]restricts to S n to exhibit S n as a double cover of RPn .That is, we have a diffeomorphismSn/ RPn .Example 18.8 (Continued). We haveSn SO(n 1)/SO(n) O(n 1)/O(n)S n is a double cover of RPnO(n) is a double cover of SO(n).This suggests trying to show thatRPn is diffeomorphic to SO(n 1)/O(n).13

Example 18.8 (Continued). Identify O(n) with its image under the embeddingO(n) SO(n 1) det AA 7 .AUsing the identificationSn/ RPnwe have a smooth transitive action of SO(n 1) on RPn .Exercise 18.9. Show that the isotropy group of [1 : 0 . . . : 0] is O(n) .Example 18.10 (Complex projective space, CPn ). CPn is the space of complex lines through the origin in Cn 1 .Equivalently,CPn Cn 1 \ {0}, where(X 0 , X 1 , . . . , X n ) λ(X 0 , X 1 , . . . , X n ), λ C \ {0}.We write the equivalence class of (X 0 , X 1 , . . . , X n ) as [X 0 : X 1 : . . . : X n ].The proof that CPn is a smooth manifold of real dimension 2n is completely analogous to the argument for realprojective space.Exercise 18.11. Show that CPn is a homogeneous manifold diffeomorphic to SU (n 1)/S(U (1) U (n)).Lecture 19[Riemannian Geometry – Lecture 19]Riemannian Geometry – Lecture 19 Isotropy Dr. Emma Carberry October 12,2015Recap from lecture 17:Definition 17.21. a Lie group G acts on a manifold M if there is a mapG M M(g, p) 7 g · psuch thate·p pfor all p Mand(gh) · p g · (h · p)g, h G, p M.for all(these force the action of each element to be a bijection)Definition 17.21 (continued). if for every g G,g: Mp 7 Mg·pis smooth then we say that G acts smoothly the isotropy subgroup Gp of G at p is the subgroup fixing p G acts transitively on M if for every p, q M there exists g G such that q g · pDefinition 17.22. A homogeneous space is a manifold M together with a smooth transitive action by a Lie groupG.14

Informally, a homogeneous space “looks the same” at every point.Since the action is transitive, the isotropy groups are all conjugateGg·p gGp g 1and for any p M we can identify the points of M with the quotient space G/Gp .Definition 17.23. A Lie subgroup H of a Lie group G is a subset of G such that the natural inclusion is animmersion and a group homomorphism (i.e. H is simultaneously a subgroup and submanifold).Since the the group action on a homogeneous space is in particular continuous, the isotropy subgroups are closedin the topology of G.Theorem 17.24 (Lie-Cartan). A closed subgroup of a Lie group G is a Lie subgroup of G.Corollary 17.25. An isotropy subgroup Gp of a Lie group G is a Lie subgroup of G.Theorem 17.26. If G is a Lie group and H a Lie subgroup then the quotient space G/H has a unique smoothstructure such that the mapG G/H G/H(g, kH) 7 gkHis smooth.We omit the proof; it is given for example in Warner, “Foundations of Differentiable Manifolds and Lie Groups”,pp 120–124.Corollary 17.27. A homogeneous space M acted upon by the Lie group G with isotropy subgroup Gp is diffeomorphic to the quotient manifold G/Gp where the latter is given the unique smooth structure of the previoustheorem.Lecture 18 continued:nExample 18.8 ( Real projective space, RP ). Recall that RPn is the space of lines through the origin in Rn 1 .Equivalently,RPn Rn 1 \ {0}, where(X 0 , X 1 , . . . , X n ) λ(X 0 , X 1 , . . . , X n ), λ R \ {0}.We write the equivalence class of (X 0 , X 1 , . . . , X n ) as [X 0 : X 1 : . . . : X n ].Example 18.8 (Continued). The natural projectionRn 1 \ {0} RPn(X 0 , X 1 , . . . , X n ) 7 [X 0 : X 1 : . . . : X n ]restricts to S n to exhibit S n as a double cover of RPn .That is, we have a diffeomorphismSn/ RPn .Example 18.8 (Continued). We haveSn SO(n 1)/SO(n) O(n 1)/O(n)S n is a double cover of RPnO(n) is a double cover of SO(n).This suggests trying to show thatRPn is diffeomorphic to SO(n 1)/O(n).15

Example 18.8 (Continued). Identify O(n) with its image under the embeddingO(n) SO(n 1) det AA 7 .AUsing the identificationSn/ RPnwe have a smooth transitive action of SO(n 1) on RPn .Exercise 18.9. Show that the isotropy group of [1 : 0 . . . : 0] is O(n) .Example 18.10 (Complex projective space, CPn ). CPn is the space of complex lines through the origin in Cn 1 .Equivalently,CPn Cn 1 \ {0}, where(X 0 , X 1 , . . . , X n ) λ(X 0 , X 1 , . . . , X n ), λ C \ {0}.We write the equivalence class of (X 0 , X 1 , . . . , X n ) as [X 0 : X 1 : . . . : X n ].The proof that CPn is a smooth manifold of real dimension 2n is completely analogous to the argument for realprojective space.Exercise 18.11. Show that CPn is a homogeneous manifold diffeomorphic to SU (n 1)/S(U (1) U (n)).Lecture 19:Isometry groupDefinition 19.1. A diffeomorphism ϕ : (M, g) (N, h) is called an isometry if ϕ (h) g, equivalently ifh(dϕp (v), dϕp (w)) g(v, w)for all p M , v, w Tp M .Definition 19.2. The set of isometries ϕ : (M, g) (M, g) forms a group, called the isometry group I of theRiemannian manifold (M, g).The isometry group is always a finite-dimensional Lie group acting smoothly on M (see eg Kobayashi, Transformation Groups in Differential Geometry, Thm II.1.2).16

Homogeneous and isotropic Riemannian manifoldsDefinition 19.3. A Riemannian manifold (M, g) is a homogeneous Riemannian manifold if its isometry groupI(M ) acts transitively on M .A homogeneous space M is just a differentiable manifold, no Riemannian metric. M is diffeomorphic to thequotient manifold G/Gp .A homogeneous Riemannian manifold also has a Riemannian metric compatible with the group action.A homogeneous Riemannian manifold “looks the same” at every point in terms both of its smooth structure and ofthe Riemannian metric.Definition 19.4. A Riemannian manifold (M, g) is isotropic at p M if the isotropy subgroup Ip acts transitivelyon the set of unit vectors of Tp M , viaIp {X Tp M hX, Xi 1} {X Tp M hX, Xi 1}(g, X) 7 dgp (X).The geometric interpretation is that the Riemannian manifold M near p looks the same in all directions.Definition 19.5. A homogeneous Riemannian manifold which is isotropic at one point must be isotropic at everypoint. Such a manifold is called a homogeneous and isotropic Riemannian manifold.A homogeneous and isotropic Riemannian manifold then looks the same at every point and in every direction: canyou think of any examples?Example 19.6. PropositionThe isometry group of S n is O(n 1). It acts transitively on S n . The isotropy group acts transitively on the unitvectors of the tangent space. Hence, S n is a homogeneous and isotropic Riemannian manifold.Example 19.6 (Continued). ProofSince O(n 1) is by definition the isometry group of Rn 1 and S n inherits its Riemannian metric from Rn 1 , theaction of SO(n 1) on S n is by isometries. Conversely, any isometry of the sphere can be extended linearly togive an isometry of Rn 1 (we will check this momentarily). Hence I(S n ) O(n 1).Example 19.6 (Continued). The proofs of the next two statements are similar.Choose any point p S n . It is unit length, so extend it to an orthonormal basis v1 p, v2 , · · · , vn 1 of Rn 1 .The matrix g with columns v1 , · · · , vn 1 is in O(n 1) and takes e1 to p. Thus, by going via e1 , the action istransitive.To see it is isotropic, we compute at e1 S n . We have seen that the isotropy subgroup is just O(n) acting on thestandard basis e2 , · · · , en 1 . Choose any unit vector v Tp S n . We need to find g such that dge1 (e2 ) v.But the differential of the linear map is itself, so we may replace dge1 with g.Again, extend v to an orthonormal basis and then we may take the columns of g to be v, v3 , · · · , vn 1 , similarlyto what we have seen before.Example 19.6 (Continued). To see that every isometry ϕ of a sphere extends to a linear map, we can write anypoint p Rn 1 as λs for λ

Riemannian Geometry Dr Emma Carberry Semester 2, 2015 Lecture 15 [Riemannian Geometry - Lecture 15]Riemannian Geometry - Lecture 15 Riemann Curvature Tensor Dr. Emma Carberry September 14, 2015 Sectional curvature Recall that as a (1;3)-tensor, the Riemann curvature endomorphism of the Levi-Civita connection ris R(X;Y)Z r X(r YZ) r Y(r XZ .