3
1 Let V be an inner product space. For a basis (v1, . . . , vn) of V, let A be the matrix of a linear transformation T : V → V in that basis. Consider also the matrices B = (hT vi, vji) and G= (hvi, vji). Prove thatAt=BG−1.
2 LetE be a rankkvector bundle over a smooth manifoldM endowed with a Riemannian metric and a compatible connection ∇E.
a. Show that if ∇E is flat, then given p ∈ M there is a neighborhood U of p and a parallel orthonormal frame s1, . . . , sk ofE defined onU.
b. Show that if, in addition, M is simply-connected, the neighborhood U can be taken to be equal toM.
3 Let f : M2 → R3 be an isometric immersion of a surface, consider the frame of vector fields
∂
∂x1, ∂x∂
2 along f and the corresponding coefficientsgij of the induced Riemannian metric.
2Globally flat normal bundle
3Normal bundle; normal connection; normal component of equation.
a. Show that the coefficients of the second fundamental form of f are given by bij = det
∂2f
∂xi∂xj, ∂f
∂x1, ∂f
∂x2
·det(gkℓ)−1/2, with respect to some choice of unit normal vector fieldξ.
b. Deduce that the Gaussian curvature
K = detAξ= det(bij) det(gij) and the mean curvature
H = trAξ= g11b22−2g12b12+g22b11
det(gij) .
4 a. Let γ : (a, b) → R3, ξ : (a, b) → S2(1) be smooth curves. A parametrized surface of the formf(u, v) =γ(u) +vξ(v) is called aruled surface. Investigate sufficient conditions for f to be an immersion. Compute that
K = −(γ′·ξ′)2
||(γ′+vξ′)×ξ||2. Deduce that the plane, cilinder and cone are flat surfaces.
b. For thehelicoid
f(u, v) = (vcosu, vsinu, au) (a >0), show that
(7.9.1) K(u, v) = −a2
(a2+v2)2
and that it is a minimal surface. Deduce its principal curvatures. It is not difficult to show that the plane and the helicoid are the only complete ruled minimal surfaces in R3.
5 a. Let γ : (a, b) → R2 be a smooth curve. A parametrized surface of the form f(u, v) = (γ1(v) cosu, γ1(v) sinu, γ2(v)), where γ1, γ2 are the components of γ, is called a surface of revolution. Show that
K= γ2′(γ1′γ2′′−γ2′γ1′′) γ1((γ1′)2+ (γ′2)2)2).
In particularK =−γ1′′/γ1 in caseγ is parametrized by arc-length.
b. For thetorus of revolution
f(u, v) = ((R+rcosv) cosu,(R+rcosv) sinu, rsinv) (R > r >0), show that
K= cosv r(R+rcosv). c. For thecatenoid
f(u, v) = (acosh(v/a) cosu, acosh(v/a) sinu, v)
(a >0), show that
(7.9.2) K(u, v) = −1
a2cosh4(v/a)
and that it is a minimal surface. It is not difficult to see that the only complete minimal sur-faces of revolution inR3are the plane and the catenoid. Interpret formulae (7.9.1) and (7.9.2) in view of exercise 2 of chapter 1.
6 LetM be a surface inR3 given as the pre-image of a regular value of a smooth mapf :U →R, whereU is an open subset ofR3. Show that the second fundamental form of M is given by
B(u, v) = 1
||(gradf)p||Hess (f)(u, v)
for some choice of unit normal vector field, where p∈M and u,v∈TpM.
7 (The Beez-Killing theorem) a. Let S,T :V → V be self-adjoint linear operators on an Euclidean vector spaceV. Suppose that rank(S) ≥3 and Λ2S = Λ2T : Λ2V → Λ2V. Prove thatS =±T.
b. Let M be a (not necessarily complete) connected Riemannian manifold of dimension n and suppose f :M → Rn+1 is an isometric immersion such that the rank of the second funda-mental form is at least 3 at every point. Prove thatf is rigid.
8 LetM ⊂N ⊂P be a chain of Riemannian submanifolds. Prove that ifM is totally geodesic in N and N is totally geodesic inP, then M is totally geodesic inP.
9 Prove that each connected component of the fixed point set of an isometry of a Riemannian manifold is a properly embedded totally geodesic submanifold. Generalize the result to the fixed point set of a group of isometries.
10 Prove that the totally geodesic submanifolds of RPn are the images of totally geodesic sub-manifolds of Sn under the projection π :Sn → RPn. Deduce that the complete totally geodesic submanifolds of RPn are isometric to RPk for some 0≤k ≤ n; in particular, the cut-locus of a point in RPn is a totally geodesic hypersurface isometric toRPn−1.
11 Consider the projection π : S2n+1\ {0} → CPn. Prove that there are exactly two kinds of complete totally geodesic submanifolds ofCPn: (i) π(V ∩S2n+1), where V is a complex subspace of Cn+1; and (ii) π(W ∩S2n+1), where W is a totally real subspace of C2n+1. Deduce that the complete totally geodesic submanifolds ofCPnare isometric toCPkor toRPkfor some 0≤k≤n;
in particular, the cut-locus of a point inCPnis a totally geodesic submanifold isometric toCPn−1. 12 Let Mn be a Riemannian submanifold of Rn+k. Fix a point p ∈ M and a normal vector ξ∈νpM. In this exercise we establish a canonical isomorphismTξ(νM)∼=TpM⊕νpM.
a. Givenu∈TpM, consider a smooth curve γ : (−ǫ, ǫ)→M with γ(0) =p, γ′(0) =u and take the parallel transport ˆξ of ξ along γ. Show that this defines a linear map TpM → Tξ(νM), and that this map is injective.
b. Given η ∈ νpM, consider the line s 7→ ξ +sη in νpM. Show that it defines a linear map νpM →Tξ(νM), and that this map is injective.
c. Show that TpM andνpM viewed as subspaces ofTξ(νM) meet only at 0. Deduce the above claim.
13 Let Mn be a Riemannian submanifold of ¯M =Rn+k. Consider the normal exponential map exp⊥:νM →Rn+k mapping ξ∈νpM top+ξ.
a. Use exercise 12 to represent the differentiald(exp⊥)ξ :TpM ⊕νpM →TpM⊕νpM as id−Aξ 0
0 id
.
b. Assume ξ is a unit vector and prove that q =p+tξ is a focal point of multiplicity m of M along the normal line through ξ if and only if 1/t is an eigenvalue of Aξ of multiplicity m.
Deduce thatdis a focal distance ofM alongξ if and only if 1/dis a principal curvature ofAξ. c. Generalize the above to other space forms to prove that: inSn+k,dis a focal distance ofM along ξ if and only if cotd is a principal curvature of Aξ; in RHn+k, d is a focal distance of M alongξ if and only if cothdis a principal curvature ofAξ.
d. In case ¯M =Sn+k, note that dis a focal distance of M along ξ if and only π−dis a focal distance ofM along −ξ.
14 (The Morse index theorem for submanifolds of Euclidean space) LetMbe a Rieman-nian submanifold of ¯M =Rn. For q∈Rn, consider the square distance function
Lq :M →R, Lq(x) = 1
2||x−q||2.
a. Prove that grad(Lq)p = (p−q)⊤. Deduce that p∈M is a critical point of Lq if and only if v=q−p∈νpM.
b. Let p ∈M be a critical point of Lq and v = q−p ∈ νpM. Prove that Hess(Lq)p =I −Av (exercise 9 of chapter 4).
c. The nullity of Lq at a critical point p is defined to be the nullity of the symmetric bilinear form Hess(Lq)p; such a critical pointpis callednon-degenerate if the nullity ofLqatpis zero.
Use Exercise 13 to deduce that the nullity of Lq at a critical point p equals the multiplicity ofq as a focal point ofM along the geodesic segmentpq. Deduce thatpis non-degenerate as a critical point ofLq if and only ifq is a non-focal point ofM along the geodesic segmentpq.
d. The index ind(Lq)p of Lq at a critical point p is defined to be the index of the symmetric bilinear form Hess(Lq)p. Show that ind(Lq)p = P
t∈(0,1)ker(I −t Av), where v = q −p.
Combine this result with part (c) to deduce that ind(Lq)p equals the sum of the multiplicities of p+tv as a focal point toM fort∈(0,1).
e. Check that this result is a specialization of the Morse index theorem 7.5.4 to the case of Euclidean submanifolds.
15 Let M be a submanifold of a Riemannian manifold ¯M. Prove that the kth-osculating space Okp(M) of M at a point p ∈ M is spanned by the k-th derivatives at 0 of all smooth curves γ : (−ǫ, ǫ) → M with γ(0) = p. (Hint: Consider the reparametrizations γ(ϑ(t)) where ϑ is a polynomial function with ϑ(0) = 0.)
16 Let M be a complete isoparametric submanifold of Euclidean space ¯M =Rn. Fix a parallel normal vector field ξ along M. Consider πξ : M → Mξ and let ˆp ∈ Mξ. Prove that the con-nected components of the level setπ−1(ˆp) are compact isoparametric submanifolds ofνpˆ(Mξ), with curvature normals given exactly by those curvature normalsvi ofM that satisfy hξ, vii= 1.
C H A P T E R 8
Isometric actions
In this chapter we extend and refine the discussion about Lie transformation groups in chapter 0.
8.1 Lie group actions
Let G be a Lie group, and let M be a smooth manifold. A left action of G on M is a smooth mapping
Φ :G×M →M satisfying the following conditions:
(a) Φ(1, p) =p;
(b) Φ(g′g, p) = Φ(g′,Φ(g, p));
for every p∈M, andg, g′ ∈G. Aright action of Gon M is defined analogously, except that one replaces condition (b) in the above definition by
(b’) Φ(g′g, p) = Φ(g,Φ(g′, p)).
One can pass from a left (resp. right) action Φ of G on M to a right (resp. left) action Ψ by setting Ψ(g, p) = Φ(g−1, p). Therefore, when working with a single action of a Lie group on a smooth manifold, it is no loss of generality to assume that this action is a left action. Most of the time we will be dealing with left actions. Right actions appear naturally in some contexts, though, and especially when we happen to have two simultaneous actions on the same manifold, one left, and one right. In any event, we make the convention that anaction ofGonM means a left action, unless explicitly stated.
Suppose Φ is an action of a Lie group G on smooth manifold M. For each g ∈ G, define a smooth map
ϕg :M →M, ϕg(p) = Φ(g, p).
Then the defining conditions of an action are equivalent to the following ones:
(c) ϕ1= idM; (d) ϕg′g =ϕg′ ◦ϕg;
whereg,g′ ∈M. It is an immediate consequence of the above that, for everyg∈G, ϕg−1 =ϕ−1g
so that ϕg is a diffeomorphism of G. Now we can say that the map g7→ϕg
is a group homomorphism fromG into the group Diff(M) of all diffeomorphisms of M. However, if one wanted to make this map into a Lie group homomorphism, then it would be necessary to
introduce a structure of infinite-dimensional smooth manifold on Diff(M), as this space is not a finite-dimensional manifold in a natural way. This indeed can be done under some assumptions on M, but we do not have interest in it, since we are considering only finite-dimensional smooth manifolds and Lie groups in this book. A short and good introductory reference about infinite-dimensional Lie groups is [Mil84].
Let g denote the Lie algebra of G. Given an action g∈G7→ ϕg ∈Diff(M), the elements of g induce smooth vector fields onM as follows. For eachX∈g, consider the associated one-parameter subgroupt7→exp(tX) of G. For eachp∈M,ϕexp(tX)(p) is a smooth curve inM; defineX∗(p) to be its tangent vector at t= 0, namely,
X∗(p) = d dt
t=0ϕexp(tX)(p).
It is clear that X∗ is a smooth vector field on M, and that its flow is given by {ϕexp(tX)}. The map X 7→ X∗ is a linear map from g into Γ(T M) which turns out to be a Lie algebra skew-homomorphism, namely:
8.1.1 Lemma With the above notations,
[X∗, Y∗] =−[X, Y]∗, for everyX, Y ∈g.
Proof. Let f be a smooth function onM, let p∈M, and consider the smooth functions F(r, s, t) =f(exp(rX) exp(sY) exp(tX)p)
and
G(s, t) =F(−t, s, t), respectively defined onR3 and R2. Then
(8.1.2)
∂2G
∂s∂t(0,0) = ∂2F
∂s∂t(0,0,0)− ∂2F
∂r∂s(0,0,0)
=Xp∗(Y∗f)−Yp∗(X∗f)
= [X∗, Y∗]p(f).
On the other hand,
exp(−tX) exp(sY) exp(tX) = exp(sAdexp(−tX)Y), so
∂G
∂s(0, t) = d ds
s=0f(exp(sAdexp(−tX)Y)p)
= (Adexp(−tX)Y)∗p(f), and
(8.1.3) ∂2G
∂t∂s(0,0) =−[X, Y]∗p(f).
Comparing (8.1.2) and (8.1.3) yields the desired result.
1
Let Φ and Ψ denote actions of the Lie groupGon the smooth manifoldsM andN, respectively.
A smooth map f :M →N is called equivariant with respect to the actions of Gon M and N if f(Φ(g, p)) = Ψ(g, f(p)),
for everyp∈M,g∈G. A subset A⊂M is called invariant if Φ(g, p)∈A for everyp∈A,g∈G.
A simplification of the notation is in order. Whenever we will be working with one single action, and there will be no possibility of confusion, we will agree to denote an action of G on M simply by (G, M)2 and we will denote ϕg(p) simply by gp. So suppose a action (G, M) is given. The orbit through a pointp∈M is the following subset ofM:
Gp={gp∈M |g∈G}; theisotropy subgroup atp∈M is the following subgroup of G:
Gp ={g∈G|gp=p}.
It is indeed obvious that Gp is a subgroup of G. Owing to the continuity of the action, it is also true that Gp is a closed subgroup of G. It follows that Gp is Lie subgroup of Gwith the induced topology (cf. Remark 0.4.7); furthermore, denoting the Lie algebra of bygp, we have
gp ={X∈g|Xp∗= 0};
and, as is very easy to check, the various isotropy subgroups at points of the same orbit are conjugate among themselves, namely,
(8.1.4) Ggp=gGpg−1,
for everyp∈M andg∈G. Next, we introduce the smooth map ωp:G→M ωp(g) =gp,
called the orbit map through p. Plainly, the image of the orbit map ωp is the orbit Gp, and ωp induces a bijection between the quotient spaceG/Gp and the orbitGp; denote byωp the quotient map:
G ωp
> M
G/G∨ p ω¯p
>
More can be said. The orbit map ωp satisfies the equation ωp◦Lg =ϕg◦ωp,
for everyg∈G. SinceLg and ϕg are diffeomorphisms, it follows thatωp has constant rank. Since d(ωp)1(X) = Xp∗, the kernel of ωp coincides with gp. Now, Gp being a closed subgroup of G, we have thatG/Gpadmits a unique structure of smooth manifold such that the projection G→G/Gp
1Mention integration problem?
2Use other symbol?
is a submersion (cf. (0.4.18)) and, as such,G/Gp is called a homogeneous space of the Lie groupG.
Since the tangent spaceT[Gp](G/Gp) is canonically isomorphic to the quotientg/gp, it follows that ωp is an immersion of G/Gp intoM. Hence,Gpacquires an structure of immersed submanifold of M. It is clear that the tangent space
(8.1.5) Tp(Gp) = imd(ωp)1={Xp∗ ∈TpM |X∈g}.
A natural question that one can pose now is to ask when the orbits of a action (G, M) will be embedded submanifolds ofM. This is equivalent to requiring that the various orbit maps be proper maps, and is an immediate consequence of the following topological assumption on the action. We say that (G, M) is a proper action if the map
̺:G×M →M×M, (g, x)7→(gx, x)
is a proper map. As is easily seen, all the isotropy subgroups of a proper action are compact. Note also that the properness of an action (G, M) is automatic ifG is a compact Lie group.
The orbits of an action (G, M) can also be thought of determining an equivalence relation R inM: two points ofM are declared to belong to the same equivalence class if and only if they lie in the same orbit. Note that R is exactly the image of the map ̺. The set of equivalence classes is called the orbit space, and is denoted by G\M. The orbit space, equipped with the quotient topology, becomes a topological space. This topology is Hausdorff if the action (G, M) is proper.
Indeed, in this case,R is closed inM×M. Letp,q ∈M be such thatGp6=Gq. Then (p, q)6∈ R, so we can find open subsets U, V of M such that (p, q) ∈ U ×V ⊂ (M ×M)\ R, whence GU and GV are disjoint neighborhoods ofGp and Gq inG\M, respectively, proving that this space is Hausdorff.