The Affine Group of a Smooth Manifold

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The Affine Group of a Smooth Manifold

Mehdi Nabil

To cite this version:

Mehdi Nabil. The Affine Group of a Smooth Manifold. Master. Morocco. 2020. �hal-03298754�

MEHDI NABIL

1. Introduction

The goal of these notes is to explore various results concerning the group of affine transformations of a smooth manifold endowed with a linear connection. Many of those results involve the isometry group of a Riemannian manifold and its relation to the group of affine transformations for the Levi-Civita connection.

Let M be an n-dimensional smooth manifold. For any x ∈ M, we call a frame onM atxany linear isomorphismRⁿ−→^' TxM, the set of such frames will be de- notedL(M)_x. Clearly, the general linear groupGL(n,R)acts naturally onL(M)_x via the map:

L(M)x×GL(n,R)−→L(M)x, (z, g)7→z◦g,

and it is not hard to see that this action is simply transitive. Now define L(M) :=

q

x∈ML(M)_x

and consider the projectionπ: L(M)−→M given byπ(L(M)x) :=x.

Proposition 1.0.1. Let M be ann-dimensional manifold. Then L(M)−→^π M is a smooth principal GL(n,R)-bundle over M called the frame bundle of T M such that for any local frame{E₁, . . . , E_n} ofT M defined on an open subsetU ⊂M,the map:

σ:U −→L(M), x7→ {E_1|x, . . . , E_n|x}, (1) is a local (smooth) section ofL(M)−→^π M. Conversely, if σ:U −→L(M)is any smooth section, then there exists a local frame {E₁, . . . , E_n} of T M over U such that σis of the form (1).

In a similar way one defines on a Riemannian manifold(M,h, i)the bundle of orthonormal frames O(M) := q_x∈MO(M)x where each O(M)x consists of linear isometries (Rⁿ,h , i0) −→^' (T_xM,h, ix). It is clear that O(M)⊂ L(M), on the other hand the orthogonal groupO(n)acts simply transitively onO(M).

Proposition 1.0.2. Let (M,h , i) be an n-dimensional Riemannian manifold.

ThenO(M)−→^π M is a smooth principalO(n)-subbundle ofL(M). If{E1, . . . , En} is any local, orthonormal frame of T M defined on an open subsetU ⊂M then the map:

σ:U −→L(M), x7→ {E1|x, . . . , En|x}, (2) is a local (smooth) section ofO(M)−→^π M. Conversely, if σ:U −→O(M) is any smooth section, then there exists a local orthonormal frame {E1, . . . , En} of T M overU such that σis of the form (2).

Date: December 12, 2020.

1

Any diffeomorphism f : M −→ M induces a principal bundle automorphism f∗: L(M)−→L(M)such that the following diagram is commutative:

L(M) L(M)

M M

f_∗

π π

f

Explicitly, for any z ∈ L(M) we have f∗(z) := Tπ(z)f ◦z. Define on L(M) the Rⁿ-valued1-formθ∈Ω¹(L(M),Rⁿ)given byθz(v) :=z⁻¹(Tzπ(v)), we call it the canonical form ofL(M). We have the following result:

Proposition 1.0.3. Let M be a smooth manifold and let θ denote the canonical form of the frame bundle L(M). If f : M −→ M is any diffeomorphism of M then f_∗ preserves θ. Conversely, if A : L(M) −→ L(M) is any fiber-preserving transformation leaving θinvariant, then A=f_∗ for somef ∈Diff(M).

Proof. Letz∈L(M)andv∈TzL(M), then:

(f^∗θ)z(v) = θf∗(z)(Tzf∗(v))

= (f_∗(z))⁻¹(T_f∗(z)π◦Tzf_∗(v))

= (f_∗z)⁻¹(Tz(π◦f_∗)(v))

= (f∗z)⁻¹(Tz(f◦π)(v))

= (f_∗z)⁻¹◦T_π(z)f◦T_zπ(v)

= z⁻¹◦T_zπ(v)

= θz(v).

Conversely, we first notice that since A : L(M) −→L(M) is fiber-preserving, the map:

f :M −→M, f(x) =π(A(z)), z∈π⁻¹(x), is a well-defined diffeomorphism ofM. Now:

(A^∗θ)z(v) =θ_A(z)(TzA(v)) =A(z)⁻¹◦T_π(z)f◦Tzπ(v),

so A will preserve θ if and only ifA(z)⁻¹◦T_π(z)f =z⁻¹ for any z ∈ P which is exactly whatA=f_∗ means.

Proposition 1.0.3 states that the morphism Diff(M) →^Ψ Aut(L(M)), f 7→ f∗

sends the group of diffeomorphisms ofM isomorphically onto the subgroup of au- tomorphisms ofL(M)preserving the canonical formθ.

Definition 1.0.1. Let Gbe a Lie group with Lie algebra g. A connection form on a principalG-bundleP −→^π M is a1-formω∈Ω¹(P,g)satisfying:

1- For any z ∈P and any A ∈g, ωz(Aez) = A where Ae is the fundamental vector field on P corresponding toA, i.e

Ae_z:= d dtt=0

z·exp(−tA).

2- For anyg∈G and anyz∈P,v∈Tz(p), (R^∗_gω)_z(v) = Ad_g−1(ω_z(v)), withRg:P −→P being the map z7→z·g.

Let now∇be a linear connection onM. A diffeomorphismf :M −→M is called anaffine transformation with respect to∇ if it satisfiesf_∗(∇XY) =∇f∗Xf_∗Y for anyX, Y ∈

χ

_{(M)wheref∗X} is the vector field onM given by:

(f_∗X)_f−1(x):= (T_xf)⁻¹(X_x).

The group of such transformations will be denotedAff(M,∇). On the other hand, we say thatX ∈

χ

_{(M) is an}affine vector field if it generates a local1-parameter group of affine transformations. We are going to prove in this paragraph that Aff(M,∇) is a Lie group of dimension ≤ n²+n when given the compact-open topology. The idea is to reinterpret the group of affine transformations as a sub- group of Aut(L(M)) via the identification provided by Proposition 1.0.3, but in order to do this we need an artifact that represents the linear connection∇ on the frame bundleL(M). It turns out that the notion of connection form is the adequate solution for this task, more precisely we have the following:

Proposition 1.0.4. LetM be a smooth manifold and∇a linear connection onM. Then there exists a unique connection formω onL(M)such that for any local sec- tionσ:={E1, . . . , En} ofL(M)defined onU,σ^∗ω = Γ whereΓ∈Ω¹(U,gl(n,R)) is given by:

∇Ei=

n

X

j=1

ΓijEj.

Conversely, any connection formωonL(M)gives rise to a linear connection∇on M by means of the previous expression.

Proposition 1.0.5. Let M be a smooth manifold, ∇ a linear connection on M and ω the connection form on L(M) corresponding to ∇. Let f :M −→M be a diffeomorphism, then:

1- f ∈Aff(M,∇) if and only iff_∗ preserves the connection formω.

2- Conversely, any fiber-preserving tranformationA: L(M)−→L(M)leaving both θandω invariant is of the formA=f_∗ for somef ∈Aff(M,∇).

Proof. Define onM the linear connection∇e by the expression:

∇eXY = (f_∗)⁻¹(∇f∗Xf_∗Y)

Fix an open subsetU ⊂M and let{E1, . . . , En}be a local frame ofT M defined on U which correponds to a local section σ:U −→L(M). Clearly {f_∗E1, . . . , f_∗En} defines a local frame ofT M overV :=f⁻¹(U)and the corresponding local section is exactlyf_∗⁻¹◦σ◦f :V −→L(M). For convenience putg:=f⁻¹ and write:

∇Ee _i=

n

X

j=1

Γe_ijE_j, ∇(f_∗E_i) =

n

X

j=1

Γ_ijf_∗E_j,

where Γ := (ee Γij)i,j ∈ Ω¹(U,gl(n,R))and Γ := (Γij)i,j ∈ Ω¹(V,gl(n,R)). On the other hand, a direct computation shows that for anyx∈U:

(e∇_EkE_i)_x=g_∗(∇_f∗Ekf_∗E_i)_x = T_g(x)f((∇_f∗Ekf_∗E_i)_g(x))

=

n

X

j=1

Γ_ij((f_∗E_k)_g(x))E_i|x

=

n

X

j=1

(g^∗Γij)x(E_k|x)E_i|x.

Therefore g^∗Γ = eΓ and thus σ^∗(g^∗ω) = eΓ. In summary we proved that g^∗ω is the (unique) connection form on L(M) defining ∇. Soe f :M −→ M is an affine transformation i.e∇e =∇ if and only ifω=f^∗ω.

Theorem 1.0.1. Let M be an n-dimensional smooth manifold with a global trivi- alization {X1, . . . , Xn} of T M. DenoteG the group of transformations preserving this trivialization, i.e diffeomorphismsf :M −→M satisfyingTxf(X_i|x) =X_i|f(x). ThenGpossesses a unique Lie group structure for the compact-open topology such that dimG≤dimM. More precisely for anyp∈M, the map:

G−→M, f 7→f(p),

is an imbedding ofGonto a closed submanifold ofM, and the submanifold structure on the image is what makesGa Lie transformation group. Moreover the Lie algebra of Gconsists of complete vector fields whose1-parameter subgroups are in G.

Theorem 1.0.2. LetM be a smoothn-dimensional manifold and∇an affine con- nection onM, thenAff(M,∇)is a Lie group for the compact-topology of dimension

≤n²+n. More precisely for anyz∈L(M)the map:

Aff(M,∇)−→L(M), f 7→f_∗(z),

is injective and its image is a closed submanifold ofL(M). The submanifold struc- ture on its image makes Aff(M,∇) a Lie transformation group. Its Lie algebra consists of complete affine vector fields onM.

Proof. First some preparation. For any x ∈ M, Recall that since GL(n,R) acts simply transitively on Px :=π⁻¹(x), the map GL(n,R)−→Px,g 7→z·g and its differentialgl(n,R)−→TzPx,A7→ _{dt t=0}^d z·exp(tA)is therefore an isomorphism. In other words for anyv∈TzPx there exists a uniqueA∈gl(n,R)such thatv=Aez. On the other hand it is clear thatTzπ(TzPx) = 0and sinceTzπ:TzP −→TxM is surjective we get thatker(Tzπ) =TzPx.

Denote ω the connection form on P := L(M) corresponding to ∇ and consider the map T P −→P ×(gl(n,R)×Rⁿ), Z 7→(p(Z), ω(Z), θ(Z)), then one easily checks that the following diagram is commutative:

T P P×(gl(n,R)×Rⁿ)

P φ

p

pr₁

i.e the previous map is a vector bundle homomorphism. In fact, this map defines a global trivialization ofT P, to see this we first notice that this map is surjective given that π: P −→M is a submersion so any v ∈Rⁿ is of the form v =θz(w) and that A = ω(A)e for any A ∈ gl(n,R). Let v ∈T_zP such that θ_z(v) = 0 and ω_z(v) = 0, then:

0 =θ_z(v) =z⁻¹(T_zπ(v)),

hence v ∈ ker(Tzπ) := TzPπ(z), on the other hand write v = Aez for some A in gl(n,R)then we get that

0 =ωz(v) =ωz(Aez) =A,

and so v= 0, consequentlyφis injective. On the other hand ifσ:U −→P is any local section one can define the map:

P_|U×(gl(n,R)×Rⁿ)−→^ψ P_|U, Aeσ(π(z))+ (Tπ(z)σ)(σπ(z)(v)),

it is clear thatψis a smooth map and one checks without difficulty thatφ◦ψ= Id and thusφis a local diffeomorphism, hence we conclude thatφis a vector bundle isomorphism. Finally, let F : P −→ P be any fiber preserving transformation leavingθandωinvariant and choosev∈Rⁿ andA∈gl(n,R)such that(z, A, v) = φ(w). Then:

φ◦TzF(w) = (F(z),(F^∗ω)z(w),(F^∗θ)z(w)) = (F(z), ωz(w), θz(w)), which means thatTzF(φ⁻¹(z, A, v)) = (F(z), A, v)and soF preserves any global frame ofT P defined by φ. By Theorem 1.0.1 we get the desired result.

The remaining part of this section will be dedicated to prove that the isometry group of a Riemannian manifold is also a Lie group by following the same footsteps of the previous paragraph.

Assume now that (M,h , i) is a Riemannian manifold and let ∇ be a metric connection onM, i.e for anyX, Y, Z∈

χ

_{(M),XhY, Zi=h∇XY, Zi+hY,∇XZi.}

Let{E1, . . . , En} be a local orthonormal frame of T M defined on an open subset U ⊂M and write

∇Ei=

n

X

j=1

ΓijEj.

Then we get that Γij = −Γji or in other terms Γ ∈ Ω¹(U,so(n)). Thus if eω is the connection form on L(M)corresponding to∇ then its restrictionω to the or- thogonal frame bundle O(M) is so(n)-valued and defines therefore a connection form onO(M)and it is in fact the only connection form onO(M)representing∇.

Conversely any connection form onO(M)admits a unique extension toL(M)and defines therefore a metric connection∇ onM.

DenoteIsom(M,h, i)the isometry group of the Riemannian manifold(M,h, i).

Proposition 1.0.6. Let (M,h , i) be a Riemannian manifold, ∇ its Levi-Civita connection andω the connection form onO(M)representing∇.

1- A diffeomorphism f :M −→M is an isometry if and only if f_∗(O(M)) = O(M).

2- If A : O(M) −→ O(M) is a fiber-preserving transformation leaving in- variant the canonical formθ ofO(M), then there exists a unique isometry f :M −→M such that A=f∗.

3- Any (principal) bundle automorphismO(M)−→O(M)leavingθinvariant, leaves ω invariant.

Proof. The first point is the definition of a Riemannian isometry, the argument for the second point is the same as in Proposition 1.0.5. For the third point, observe that since ∇ is torsion-free, then ω is torsion-free as well i.e the 2-form T ∈Ω²(O(M),Rⁿ), called torsion form ofω, given by:

T :=ω∧θ+dθ, (3)

vanishes. Let A : O(M) −→ O(M) be a bundle automorphism, then A^∗ω is a connection form on O(M). Since A preserves the canonical form θ, we get from expression (3) thatA^∗ω is torsion-free as well, so by uniqueness of the Levi-Civita connection we conclude thatA^∗ω=ω.

Theorem 1.0.3. Let (M,h , i) be a Riemannian manifold, then Isom(M,h , i) with the compact-open topology is a Lie group of dimension≤ ⁿ⁽ⁿ⁺¹⁾₂ . In fact for any z∈O(M), the map:

Isom(M,h, i)−→O(M), f 7→f∗(z),

is an imbedding and its image is a closed submanifold of O(M). If ∇ is the Levi- Civita connection of h , i then Isom(M,h, i) is a closed subgroup of Aff(M,∇).

Its Lie algebra consists of complete Killing vector fields on M.

Proposition 1.0.7. Let (M,h , i) be a Riemannian manifold, then the natural action of Isom(M,h , i) on M is proper. In particular if M is compact then Isom(M,h, i)is compact.

Proof. Choose a compactK⊂M and putGK ={f ∈Isom(M), f(K)∩K6=∅}.

Let(fn)n be an arbitrary sequence ofGK, then for anyn∈Nwe can findpn∈K such thatfn(pn)∈ K. Since K is compact we can assume without loss of gener- ality that (fn(pn))n is convergent and we denote q ∈ K its limit, similarly there exists a subsequence (p_ϕ(n))n of (pn)n converging to some p ∈ K and therefore (f_ϕ(n)(p_ϕ(n)))n is also convergent.

Denote d:M ×M −→ R⁺ the geodesic distance, which is preserved by elements ofIsom(M,h, i), hence:

d(fϕ(n)(p), q) ≤ d(fϕ(n)(p), fϕ(n)(pϕ(n))) +d(fϕ(n)(pϕ(n)), q)

≤ d(p, p_ϕ(n)) +d(f_ϕ(n)(p_ϕ(n)), q) −→

n→+∞0.

Next put p = π(z) with z ∈ O(M) and C = {fϕ(n)(p), n ∈ N} ∪ {q}, then C is a compact subset of M and since O(n) is compact we get that π⁻¹(C)is a compact subset of O(M) containing the sequence ((f_ϕ(n))_∗(z))_n and so it has a convergent subsequence ((f_ψ(n))_∗(z))_n. Theorem 1.0.3 implies that {f∗(z), f ∈ Isom(M)} is closed submanifold inO(M)so ((f_ψ(n))_∗(z))_n converges tof_∗(z)for some f ∈Isom(M,h, i)and by the same result we get that(f_ψ(n))_n converges to f in Isom(M,h, i). We conclude that G_K is compact and soIsom(M,h, i)acts properly onM.

2. Results on the dimension of the affine group of a manifold In this section we state some results that illustrates how the dimension of the group of affine transformations affects the global structure of the manifold. The main paragraph of this section relies on the notion of parallel transport along curves and its how it is perceived from the perspective of the frame bundle, and this will be our starting point.

Let M be an n-dimensional smooth manifold and ∇ a linear connection on M. Letγ: [0,1]−→M be a smooth a curve, then∇provides a unique linear operator denotedDγ˙ : Γ(γ⁻¹T M)−→Γ(γ⁻¹T M)satisfying:

D_γ˙(f V) =f⁰V +f D_γ˙V, f ∈ C^∞([0,1],R), V ∈Γ(γ⁻¹T M),

we call it thecovariant derivation alongγ, here Γ(γ⁻¹T M)is the space of vector fields along γ i.e smooth maps V : [0,1]−→ T M such thatV(t) ∈T_γ(t)M. In a local frame{E1, . . . , En}where∇Ei=Pn

j=1ΓijEj we get that:

D_γ˙V(t) =

n

X

k=1

f_k⁰(t) +

n

X

i,j=1

f_i(t)g_j(t)Γ^k_ji(γ(t))

E_k|γ(t), (4)

where V(t) = Pf_i(t)E_i|γ(t) and γ(t) =˙ Pg_i(t)E_i|γ(t). We say thatV is parallel alongγifD_γ˙V = 0, using that equation (4) is a linear system of ordinary differential equations one gets the following important result:

Theorem 2.0.1. LetM be ann-dimensional manifold with a linear connection∇ and γ : [0,1]−→ M a smooth curve. For any v ∈ Tγ(0)M, there exists a unique parallel vector fieldV along γsatisfyingV(0) =v. We callV the parallel transport of v alongγ.

Letωbe the connection form onL(M)representing∇, for any basis{e1, . . . , en} of Tγ(0)M one obtain a parallel frame {E1, . . . , En} along γ i.e Ei is the par- allel vector field along γ satisfying Ei(0) = ei, and it is an easy matter to see that {E1(t), . . . , En(t)} is a basis of Tγ(t)M, thus one obtains a smooth curve eγ: [0,1]−→L(M)given by:

eγ(t) = (E₁(t), . . . , E_n(t)),

such a curve is calledthe horizontal lift ofγtoL(M)through{e1, . . . , en}.

Proposition 2.0.1. Let M be ann-dimensional manifold with a linear connection

∇ and letγ : [0,1]−→ M and α: [0,1] −→ L(M) be smooth curves. Then α is a horizontal lift of γ if and only ifπ◦α=γ and ω_α(t)( ˙α(t)) = 0, where ω is the connection form corresponding to ∇.

Proof. Sinceπ(α(s)) =γ(s)for any0≤s≤1, we can writeα(s) = (V₁(s), . . . , V_n(s)) where V_i ∈Γ(γ⁻¹T M). Next choose a local frame {E₁, . . . , E_n} ofM defined on an open subsetU ⊂M and let >0 such thatγ(]t−, t+[)⊂U then write:

Vi(s) =

n

X

j=1

gij(s)Ej|γ(s), ∇Ei=

n

X

j=1

ΓijEj, (5)

fors∈]t−, t+[. Then it is straightforward to prove that:

Dγ˙Vi(t) =

g⁰(t) +g(t)·Γ_γ(t)( ˙γ(t))

·Ej|γ(t),

where g := (gij)i,j ∈ C^∞([0,1],GL(n,R))and Γ := (Γij)i,j. On the other hand if σ:U −→L(M)is the local section corresponding to{E1, . . . , En} then (5) means exactly thatα(s) =σ◦γ(s)·g(s)⁻¹, for convenience denoteg(s) :=ˆ g(s)⁻¹. Thus:

˙

α(t) = T_(σ◦γ)(t)R_g(t)((σ◦γ)⁰(t)) + d ds|s=t

(σ◦γ)(t)·g(s)ˆ

= T_(σ◦γ)(t)R_g(t)((σ◦γ)⁰(t)) + d ds|s=t

(σ◦γ)(t)·g(t)ˆ ·(g(t)ˆg(s))

| {z }

h(s)

.

Nows7→h(s)is a curve inGsatisfyingh(t) =eand therefore:

d ds|s=t

(σ◦γ)(t)·ˆg(t)·h(s) =−hg⁰(t)_{(σ◦γ)(t)·ˆg(t)},

where hg⁰(t) is the fundamental vector field on L(M) corresponding to −h⁰(t) =

−g(t)ˆg⁰(t). Thus using that R^∗_gω = Ad_g−1(ω) and ω(A) =e A we get from the previous computation:

ω_α(t)( ˙α(t)) = g(t)·(σ^∗ω)_γ(t)( ˙γ(t))·g(t)⁻¹−g(t)ˆg⁰(t)

= g(t)·Γ_γ(t)( ˙γ(t))·g(t)⁻¹−g(t)ˆg⁰(t)

= g(t)·Γ_γ(t)( ˙γ(t))·g(t)⁻¹−g⁰(t)ˆg(t)⁻¹ (g⁰(s)ˆg(s) = In)

=

g(t)·Γγ(t)( ˙γ(t))−g⁰(t)

·ˆg(t)⁻¹

So we conclude thatω_α(t)( ˙α(t)) = 0if and only if Dγ˙Vi(t) = 0for all1≤i≤n.

Recall that a vector fieldZonL(M)is calledhorizontalifω(Z) = 0andstandard ifθ(Z)is a constant function.

Proposition 2.0.2. Let M be ann-dimensional manifold with a linear connection

∇ andω the connection form of∇ onL(M).

(1) Let Z be a standard horizontal vector field on L(M). For any z ∈L(M), the curve defined byγ(t) :=π(ϕ^Z_t(z))is a geodesic on M.

(2) Conversely, given a geodesicγ: [−a, a]−→M, there exists a local standard horizontal vector field Z on L(M) and > 0 such that γ(t) = ϕ^Z_t(z) for any − < t < .

Proof. Letz∈L(M)then there exists >0such that the curveα:]−, [−→L(M) given by α(t) = ϕ^Z_t(z) is well-defined and smooth. Define γ(t) = π(α(t)), since ω_α(t)(α⁰(t)) =ω_α(t)(Z_α(t)) = 0then Proposition 2.0.1 gives that αis a horizontal lift ofγonL(M), therefore if we write:

α(t) = (V₁(t), . . . , V_n(t)), V_i∈Γ(γ⁻¹T M),

we get that{V1, . . . , Vn} is a parallel frame alongγ. On the other hand if we write θz(Zz) = (a1, . . . , an)then we get that:

γ⁰(t) =T_α(t)π(Z_α(t)) =α(t)(θ_α(t)(Z_α(t))) =α(t)(θz(Zz)) =

n

X

i=1

aiEi(t), which shows thatγ⁰ is parallel alongγi.eγ is a geodesic.

Conversely, letγ: [−a, a]−→M be a geodesic and letU be a normal neighborhood ofp:=γ(0)inM, i.e there exists an open neighborhoodV of0in T_pM such that exp_p:V −→U is a diffeomorphism. For anyx∈U, there exists a unique normal

geodesic t 7→ (exp_p(texp⁻¹_p (x))) joining p to x, denote αz : [0,1] −→ L(M) its (unique) horizontal lift throughz∈L(M)p and define:

Zα_z(1):=α⁰_z(1).

It is clear from this definition thatωz(Zz) = 0and by uniqueness of the horizontal lift, one proves thatα⁰_z(t) =Z_αz(t). Furthermore be shown thatZdefines a smooth vector field on a neighborhood of anyz ∈L(M)_γ(0) in which caseα_z(t) =ϕ^Z_t(z).

Since γ is a normal geodesic in a small neighborhood ]−, [, we get the desired result.

Denoteaff(M,∇) := Lie(Aff(M,∇)), anyX ∈aff(M,∇)defines a smooth vector fieldXˆ ofL(M)given by:

Xˆz:= d dtt=0

exp(tX)∗(z).

It is clear thatXˆ is a complete vector field onL(M).

Proposition 2.0.3. Let M be an n-dimensional manifold, ∇ a linear connection onM, and letX ∈aff(M,∇). Suppose thatωz( ˆXz) = 0for somez∈L(M). Then the curve γ : R −→ M, γ(t) = exp(tX)·x with x = π(z) is a geodesic and its horizontal lift atz is the curveγ(t) := exp(tX)ˆ ∗(z),t∈R.

Proof. Put ˆγ(t) = exp(tX)_∗(z), then clearly π(ˆγ(t)) = γ(t), moreover from the relation exp(tX)^∗ω = ω we get thatωexp(tX)_∗z( ˆXexp(tX)_∗z) = ωz(Zz) = 0 which means thatˆγ is the horizontal lift ofγ throughz, in particular if z= (e1, . . . , en) then:

ˆ

γ(t) = (E1(t), . . . , En(t)),

Ei being the parallel transport ofei alongγ. Moreoverexp(tX)^∗θ =θgives that θ_exp(tX)∗z( ˆX_exp(tX)∗z) =θ_z( ˆX_z)so if we put θ_z(X_z) = (a₁, . . . , a_n)we get that:

γ⁰(t) =T_γ(t)ˆ π( ˆX_ˆγ(t)) = ˆγ(t)(θ_γ(t)ˆ ( ˆX_ˆγ(t))) = ˆγ(t)(θ_z( ˆX_z)) =

n

X

i=1

a_iE_i(t).

Henceγ⁰ is parallel alongγ, i.eγis a geodesic.

Theorem 2.0.2. LetM be ann-dimensional manifold with a linear connection∇.

Thendim(Aff(M,∇)) =n(n+ 1)if and only ifM is an ordinary affine space with the natural flat affine connection.

Proof. DenoteG:= Aff(M,∇)and for anyx∈M denoteGx the isotropy atxfor the natural action ofGonM. First note that the map:

G_x−→GL(T_xM), f 7→T_xf (6) is an injective Lie group homomorphism. Assume now thatdimG=n(n+ 1), then letx∈M andz∈L(M)such that π(z) =x. Since the map:

G−→^Ψ L(M), f 7→f_∗(z)

is an imbedding ofGonto a closed submanifold ofL(M)anddim L(M) =n(n+ 1), then eitherΨ(G) = L(M)orΨ(G)is a connected component ofL(M)and in any case we get thatM =G·x'G/Gx, therefore:

dimG_x= dimG−dimM =n².

This gives in particular thatG⁰_x = GL⁺(TxM)under the identification (6). Now lett > 0 and consider the transformation At∈ GL⁺(TxM)given by At(u) =tu.

From the previous remark there existsft∈G⁰_xsuch thatTxft=At, hence for any u, v, w∈TxM we get that:

A_t(R^∇_x(u, v)w) =R_x^∇(A_tu, A_tv)A_tw, A_t(T_x^∇(u, v)) =T_x^∇(A_tu, A_tv), thereforeR_x^∇(u, v)w=t⁻²R^∇_x(u, v)wandT_x^∇(u, v) =t⁻¹T_x^∇(u, v)for allt >0, and so we conclude thatR^∇= 0 andT^∇= 0.

On the other hand, letZ be a standard horizontal vector field onL(M)i.eω(Z) = 0 and θ(Z) is constant whereω is the connection form representing ∇ andθ is the canonical form ofL(M). Ifg:= Lie(G)then there exists a uniqueX ∈gsuch that Zz= ˆXz where:

Xˆ

ez:= d dtt=0

exp(tX)_∗ez, ze∈L(M).

From Proposition 2.0.3 we get thatγ(t) = exp(tX)·xis a geodesic with horizontal lift at z the curveγ(t) = exp(tX)ˆ ∗(z)defined for any t∈R. Now γ:R−→M is the geodesic with initial conditionsγ(0) =xandγ⁰(0) =T_zπ(Z_z)and therefore its horizontal lift atz is exactly

α:]−, [−→L(M), t7→ϕ^Z_t(z),

which proves thatαcan be extended to all ofR. Sincez∈L(M)was arbitrary we get thatZis complete, and since we know by Proposition 2.0.2 that geodesics ofM are exactly the projections of integral curves of standard horizontal vector fields, we conclude thatM is (geodesically) complete.

Consider now the universal cover Mf of (M,∇) with its induced induced linear connection, then there exists an affine transformationMf−→^' Rⁿ. NextM =M /Γf where Γ is a discrete subgroup of Aff(M ,f ∇)'GL(n,R)o Rⁿ hence commuting with Aff(M ,f ∇)⁰ ' GL⁺(n,R)o Rⁿ. But one can show with ease that only the trivial element commutes with connected component ofGL(n,R)o Rⁿ, henceΓis trivial andM is itself simply connected i.eMf=M, this completes the proof.

Theorem 2.0.3. Let M be an n-dimensional manifold with an affine connection and assume thatdim Aff(M,∇)> n². Then∇ is torsion-free.

This result is a consequence of the following algebraic Lemma:

Lemma 2.0.1. Let V ba an n-dimensional vector space and T :V ×V −→V a non-trivial skew-symmetric bilinear map i.eT ∈V⊗Λ²V^∗. DenoteH the subgroup of linear transformation preservingT, thendimH ≤n²−n.

Proof of the Theorem. Denote G:= Aff(M,∇)then letx∈M and denote Gxthe isotropy atxfor the natural action ofGonM, then fromG/Gx'G·xwe get:

dim(Gx)≥dim(G)−dim(M)> n²−n (7) On the other hand denoteT^∇ the torsion tensor of∇, then for every f ∈G_x we get that:

T_x^∇(T_xf(u), T_xf(v)) =T_xf(T_x^∇(u, v)), u, v ∈T_xM.

Therefore the group {Txf, f ∈ Gx} ' Gx preserves T_x^∇, but according to the previous Lemma and (7) we conclude thatT_x^∇= 0for anyx∈M, i.e∇is torsion- free.

Another result in the same spirit is the following Theorem due to Egorov [3] and can be proved by essentially the same procedure:

Theorem 2.0.4. LetM be ann-dimensional manifold and ∇ a linear connection on M such that dim Aff(M,∇)> n². Then ∇ has neither torsion nor curvature provided that n≥4.

3. Relation between the isometry group and the affine group of a Riemannian manifold

In this section, we go through a number of results that have treated the relation between the group of affine transformations and the isometry group in the case of a Riemannian manifold. These results exploit the holonomy Riemannian manifolds along with the De Rham decomposition theorem in order to get to conclusions, therefore we begin by introducing the notion of holonomy.

Let M be an n-dimensional manifold and ∇ a linear connection on M. For any smooth curve γ: [a, b]−→M one can define a mapτ^γ_a,b :Tγ(a)M −→ Tγ(b)M by the formula τ^γ_a,b(v) = V(b) where V ∈ Γ(γ⁻¹T M) is the parallel transport of v alongγ(relative to∇). It is clear that this map is linear sinceV is a solution of a system of linear ordinary differential equations, furthermore:

Proposition 3.0.1.

(1) τ^γ_a,b does not depend on the orientation-preserving parametrization of the curveγ.

(2) Denoteγ₁:=γ_|[a,t0] andγ₂:=γ_|[t0,b] i.eγ=γ₁∗γ₂, then:

τ^γ_a,b=τ^γ_t²

0,b◦τ^γ_a,t¹₀.

(3) For any smooth curve γ,τ^γ_a,b is an isomorphism and its inverse is exactly the linear operator τ^γ_a,b⁻ :T_γ(b)M −→τ_γ(a)M withγ⁻(t) =γ(a+b−t).

Proof.

(1) & (3)- Choose a diffeomorphism ϕ: [c, d]−→[a, b] and put α=γ◦ϕ. Let V be the parallel transport ofv∈T_γ(a)M alongγandW :=V ◦ϕ. Next, write:

V(ϕ(s)) =

n

X

i=1

g_i(ϕ(s))E_i|α(s), γ(ϕ(s)) =˙

n

X

i=1

f_i(ϕ(s))E_i|α(s)

for any s ∈]t−, t+[, where {E1, . . . , En} is a local frame of M defined on an open subsetU andα(]t−, t+[)⊂U. Sinceα(s) =˙ ϕ⁰(s) ˙γ(ϕ(s)), then:

Dα˙W(s) =

n

X

i=1

ϕ⁰(s)g⁰_i(ϕ(s))Ei|α(s)+

n

X

i=1

gi(ϕ(s))(∇α˙Ei)_|α(s)

=

n

X

i=1

ϕ⁰(s)g⁰_i(ϕ(s))E_i|α(s)+

n

X

i,j=1

ϕ⁰(s)gi(ϕ(s))fj(ϕ(s))(∇E_jEi)_|α(s)

= ϕ⁰(s)(Dγ˙V)(ϕ(s))

= 0

Thus D_α˙W(t) = 0 for any t ∈ [c, d] i.e W is parallel along α, moreover if ϕ is orientation-preserving we get thatW is the parallel transport ofv alongα, thus:

τ^α_c,d(v) =W(d) =V(b) =τ^γ_a,b(v).

In the case where[c, d] = [a, b] andϕ(t) =a+b−t i.eα=γ⁻ then we get that:

τ^γ_a,b⁻◦τ^γ_a,b(v) =τ^γ_a,b⁻(V(b)) =τ^γ_a,b⁻(W(a)) =W(b) =V(a) =v.

(2)− Let V be the parallel transport of v ∈ Tγ(a)M along γ and denote V1 the parallel vector field along γ₁ satisfyingV1(a) = v and V2 the parallel vector field along γ₂ with initial condition V2(t0) = V1(t0). Then by uniqueness of parallel transport and the expressions ofγ₁ and γ₂, we getV₁ =V_|[a,t0] and V₂ :=V_[t0,b]. In particular:

τ^γ_t²

0,b◦τ^γ_a,t¹₀(v) =τ^γ_t²

0,b(V(t0)) =V(b) =τ^γ_a,b(v).

This ends the proof.

These properties allows to extend the definition of τ^γ_a,b for piecewise smooth curvesγ: [a, b]−→M by setting:

τ^γ_a,b :=τ^γ_t

k,b◦τ^γ_tk−1,t

k◦ · · · ◦τ^γ_t

1,t₂◦τ^γ_a,t

1,

where a=t0< t1 <· · · < tk < tk+1 =b is any subdivision of[a, b]such that the curveγ_[ti,ti+1] is smooth. The previous properties extend to this situation as well:

Proposition 3.0.2. LetM be a smooth manifold with a linear connection∇. Then:

(1) τ^γ_a,b does not depend on the orientation-preserving parametrization of the piecewise smooth curveγ: [a, b]−→M.

(2) Given two piecewise smooth curves γ₁: [a, b]−→M and γ₂ : [b, c]−→M such thatγ₂(b) =γ₁(b)then τ^γa,c^1∗γ2=τ^γ_b,c² ◦τ^γ_a,b¹.

(3) For any piecewise smooth curve γ : [a, b] −→ M, τ^γ_a,b is invertible with inverseτ^γ_a,b⁻.

It is more convenient therefore to denote τ^γ_a,b byτ^γ_γ(a),γ(b) instead or justτ^γ_γ(a) whenγ is a loop. Fixx0∈M and define:

Holx₀(M,∇) :={τ^γ_x0 :Tx₀M −→^' Tx₀M, γis a loop based atx0}.

Using the above observations, it is clear thatHol_x0(M,∇)is a subgroup ofGL(T_x0M) called theholonomy group of (M,∇)atx₀. We also define therestricted holonomy

of (M,∇)atx0 to be:

Holg_x0(M,∇) :={τ^γ_x

0 :T_x0M −→^' T_x0M, γis a contractible loop based atx₀}, which is obviously a subgroup of the holonomy group since concatenation and in- verse of contractible loops remains contractible. It is also straightforward to see thatHolgx₀(M,∇)is normal inHolx₀(M,∇).

Theorem 3.0.1. Let M be a smooth manifold and ∇ a linear connection on M. The holonomy group Hol_x0(M,∇) possesses the structure of an (immersed) Lie subgroup ofGL(Tx₀M)andHolgx₀(M,∇) = Holx₀(M,∇)⁰.

Proof.

We start by proving thatgHolx₀(M,∇)is arcwise connected. Letg∈gHolx₀(M,∇), then there exists a piecewise smooth loopγbased atx0 homotopic to the constant loopx0 such thatg=τ^γ_x0. DenoteF : [0,1]×[a, b]−→M this homotopy, i.e:

F(0, t) =γ(t), F(1, t) =x0, F(s, a) =F(s, b) =x0.

PutF(s, t) :=γ_s(t), clearly each loopγ_sis contractible as well and one can assume without loss of generality thatγ_sis piecewise smooth for any0≤s≤1, so we get a correspondence α: [0,1]−→gHol_x0(M,∇), s7→τ^γ_a,b^s withα(0) =g and α(1) = Id.

The claim amounts to proving thatαis continuous.

Fix a system of local coordinates(U, x¹, . . . , xⁿ)centered atx0. For every0≤s≤1, F(s,1) =x0∈U therefore we can find an open neighborhoodUs ofsin [0,1]and 0≤α_s<1such thatF(U_s×[α_s,1])⊂U. Since[0,1]is compact, it is the union of finitely manyU_si with1≤i≤r, so if we putα= max

1≤i≤rα_si <1we obtain that:

F([0,1]×[α,1]) =F[^r

i=1

Us_i

×[α,1]

=

r

[

i=1

F(Us_i×[α,1])⊂U.

A similar argument allows to choose α <1such that every γ_s is smooth on[α,1].

Now letv∈T_x0M and denoteV_sthe parallel transport ofvalongγ_s. PutE_i:= _∂x^∂_i and write for anyα≤t≤1:

Vs(t) =

n

X

i=1

fi(s, t)E_i|F(s,t), ∂F

∂t(s, t) =

n

X

i=1

gi(s, t)E_i|F(s,t), ∇Ei=

n

X

j=1

ΓijEj. Then Vs|[α,1] is the solution of the following system of ordinary linear differential equations:

n

X

k=1

∂f_k

∂t (s, t) +

n

X

i,j=1

g_j(s, t)f_i(s, t)Γ^k_ji(F(s, t))

E_k|F(s,t)

It is clear that the coefficients of this system are all continuous (in fact uniformly continuous) with respect to the parameter s, therefore we get that the solution [0,1]×[α,1]−→ Rⁿ, (s, t)7→ (f₁(s, t), . . . , f_n(s, t)) is continuous as well, in par- ticular one obtains that the map [0,1] −→ T_x0M, s 7→ V_s(1) = α(s)(v) is also continuous for everyv∈T_x0M. SinceT_x0M is a finite-dimensional vector space we conclude thatα: [0,1]−→Holgx0(M,∇)is continuous.

By Yamabe’s Theorem we get that the restricted holonomy group gHol_x0(M,∇)

has the structure of an (immersed) Lie subgroup ofGL(Tx₀M). Next, observe that there exists a well-defined group homomorphism:

π₁(M, x₀)−→Hol_x0(M,∇)/gHol_x0(M,∇), [γ]7→[τ^γ_x

0],

proving that the quotient Holx0(M,∇)/gHolx0(M,∇)is at most countable. Using this we can show thatHolgx₀(M,∇)is the identity component ofHolx₀(M,∇), and sinceHolx₀(M,∇)is second-countable with countably many connected components, it has the structure of an (immersed) Lie subgroup ofGL(Tx₀M).

IfM is connected, the holonomy group at any point is essentially the same, in the sense of the following result:

Proposition 3.0.3. Let M be a connected manifold with a linear connection ∇ and let x, y∈M. Then for any piecewise smooth curveγ : [a, b]−→M joiningx toy, the map:

Hol_x(M,∇)−→Hol_y(M,∇), g7→τ^γ_x,y◦g◦(τ^γ_x,y)⁻¹, is an isomorphism.

Proof. Straightforward

In what follows(M,h, i)will always be a connected Riemannian manifold with Levi-Civita connection ∇. Then following that ∇h , i = 0 we get that for any smooth curveγ: [a, b]−→M and any V, W ∈Γ(γ⁻¹T M):

d

dthV(t), W(t)i=hD_γ˙V(t), W(t)i+hV(t), D_γ˙W(t)i.

In particular if V andW are parallel alongγ then t7→ hV(t), W(t)iis a constant map and thereforehτ^γ_a,b(v), τ^γ_a,b(w)i=hv, wi. This leads to the following result:

Proposition 3.0.4. Let (M,h , i) be a Riemannian manifold with Levi-Civita connection∇. ThenHol_x(M,∇)⊂O(T_xM,h, i)for anyx∈M.

We use the symbolHol_x(M,h, i)instead ofHol_x(M,∇)in what follows. For any x∈M, we will say thatTxM is irreducibe if it is irreducible as a Holx(M,h, i)- module i.e does not admit any proper, non-trival subspace that is invariant by the action of Holx(M,h , i). In view of Proposition 3.0.3 we see that ifTxM is irreducible thenTyM is also irreducible for anyy∈M. This suggests the following definition:

Definition 3.0.1. A Riemannian manifold (M,h , i) is said to be irreducible if TxM is irreducible for some (hence every)x∈M.

The next result is a classical Theorem in riemannian geometry and will be es- sential for the next part:

Theorem 3.0.2(De Rham decomposition theorem). A simply connected, complete Riemannian manifold (M,h, i) is isometric to the direct product M0×. . . ,×Mk

whereM0is a Euclidean space andM1, . . . , Mkare all simply connected, irreducible Riemannian manifolds. Such a decomposition is a unique up to an order of the factors involved.

Corollary 3.0.1. Let(M,h, i)be a simply connected, complete Riemannian man- ifold and M =M₀× · · · ×M_k its de Rham decomposition. Let x= (x₀, . . . , x_k), then:

(1) The identificationHolx₁(M1,h, i)×. . .Holx_k(Mk,h, i)7→Holx(M,h, i) given by(τ^γx¹₁, . . . , τ^γx^k_k)7→τ^α_x¹◦ · · · ◦τ^α_x^k is an isomorphism, where αi is the loop given by:

αi(t) = (x1, . . . , γ_i(t), . . . , xk).

(2) Under the previous identification, Hol_xi(M_i,h, i)is a normal subgroup of Hol_x(M,h, i)acting trivially onT_xjM_j forj6=iand irreducibly onT_xiM_i. (3) For anyf ∈Aff(M,∇)and any i= 1, . . . , k,

T_xf(T_x0M₀) =T_f(x)0M₀, and T_xf(T_xiM_i) =T_f(x)jM_j,

for somej= 1, . . . , k. Iff ∈Aff(M,∇)⁰ then T_xf(T_xiM_i) =T_f(x)iM_i. Proof. The first and second point are a direct consequence of the De Rham decom- position, we only prove the third point. Letf :M −→M be an affine transforma- tion and choose a piecewise smooth loopγ : [0,1]−→M based at f(x). Then for anyv∈Tx₀M0:

τ^γ_f(x)(Txf(v)) =Txf(τ^f_x^−1◦γ(v)) =Txf(v),

thusTxf(v)is invariant byHol_f(x)(M,∇)which gives thatTxf(Tx₀M0) =T_f(x)0M0. On the other hand, ifw∈Tx_iMi fori6= 0then:

τ^γ_f(x)(Txf(w)) =Txf(τ^f_x^−1◦γ(w))∈Txf(Tx_iMi),

which shows thatTxf(Tx_iMi)is invariant, furthermore ifV ⊂Txf(Tx_iMi)is any in- variant subspace then in the same way(Txf)⁻¹(V)is an invariant subspace ofTx_iMi

thus it is either trivial or equal to Tx_iMi proving that Txf(Tx_iMi) is irreducible, in particular one gets the decomposition of Tf(x)M into the sum of irreducible subspaces:

T_f(x)M =Txf(Tx₀M0)⊕ · · · ⊕Txf(Tx_kMk),

and by uniqueness of such decomposition we conclude thatTxf(Tx_iMi) =T_f(x)jMj

for somej= 1, . . . , k.

Next let X be a complete affine vector field on M i.e X ∈ aff(M,∇), and con- sider the curveγ(t) = exp(tX)·x. Letvi∈Tx_iMi, and consideru:R−→Rgiven by:

u(t) =hTxexp(tX)(vi), τ^γ_0,t(vi)i_γ(t).

Thenuis a smooth function satisfying u(0) =hvi, viix6= 0and thereforeu(t)6= 0 for−δ < t < δ, which shows thatTxexp(tX)(vi)∈T_γ(t)iMi for all−δ < t < δ. In fact sinceTx_iMi is finite-dimensional, one can chooseδ >0 small enough so that

Txexp(tX)(Tx_iMi)∈T_γ(t)iMi,

for any−δ < t < δ. The result follows from the fact thatAff(M,∇)⁰ is generated by1-parameter subgroups.

Theorem 3.0.3. Let M = M0× · · · ×Mk be the de Rham decomposition of a complete, simply connected Riemannian manifold(M,h, i).Then:

Isom(M,h, i)⁰= Isom(M₀,h, i)⁰× · · · ×Isom(M_k,h, i)⁰, Aff(M,∇)⁰= Aff(M₀,∇)⁰× · · · ×Aff(M_k,∇)⁰, where∇ is the Levi-Civita connection of M.

Proof. Consider the homomorphism Ψ : Diff(M0)× · · · ×Diff(Mk) −→ Diff(M) which corresponds to anyk-tuple of diffeomorphisms(f0, . . . , fk)the transformation f :M −→M given by:

f(x₀, . . . , x_k) = (f₀(x₀), . . . , f_k(x_k)).

ClearlyΨis continuous and injective. We claim thatf = Ψ(f0, . . . , fk)is an affine transformation if and only iffiis an affine transformation for any0≤i≤k. Indeed letγ: [0,1]−→M be any piecewise smooth curve and writeγ:= (γ₀, . . . , γ_k)then choosev=v0⊕ · · · ⊕vk ∈T_γ(0)M withvi∈T_γi(0)Mi, then:

T_γ(1)f◦τ^γ_0,1(v) =

k

X

i=1

Ty_ifi(τ^γ_0,1ⁱ(vi)), τ^f◦γ_0,1 (T_γ(0)f(v)) =

k

X

i=1

τ^f_0,1^i◦γi(Tx_ifi(vi)), which shows thatf preserves parallel transports onM if and only if eachfidoes so onMiproving the claim. One can also prove in a similar way thatf is an isometry if and only if everyfi is an isometry. In particular:

Ψ(Aff(M₀)×· · ·×Aff(Mk))⊂Aff(M), Ψ(Isom(M₀)×· · ·×Isom(Mk))⊂Isom(M).

Letf ∈Aff(M,∇)⁰ andpr_i :M −→M_i be the projection on thei-th component then denote gi := pr_i ◦f, we will show that gi(x0, . . . , xk) only depends on xi. Indeed let x= (x0, . . . , xk)∈M, j 6=i andvj ∈ Tx_jMj then by(3) of Corollary 3.0.1:

T_xg_i(v_j) =T_f(x)pr_i(T_xf(v_j)

| {z }

∈M_j

) = 0

Therefore if we fix(a0, . . . , ak)∈M and definefi:Mi−→Mi by the expession:

f_i(y) :=g_i(a₀, . . . , y, . . . , a_k),

thenf_iis a well-defined diffeomorphism ofM_iandf = Ψ(f₀, . . . , f_k). It also follows that iff ∈Isom(M,h, i)⁰then eachf_i is an isometry.

Theorem 3.0.4. Let (M,h , i) be a complete, irreducible Riemannian manifold, then Aff(M,∇) = Isom(M,h , i) except when M is a 1-dimensional Euclidean space.

The proof of this Theorem will be done in two steps: First one proves that on any such manifold, any affine transformation is homothetic and if furthermore(M,h, i) is not Euclidean then homothetic transformations are isometries, the result follows then by observing that only1-dimensional Euclidean spaces can be irreducible.

Let (M,h , i) be a Riemannian manifold, and recall that f ∈ Diff(M) is said to be ahomothetic transformation if there exists a positive constantc >0 such that hTxf(v), T_xf(w)i=c²hv, wifor allx∈M andv, w∈T_xM, i.ef^∗h, i=c²h, i. It is a well-known fact that if∇ is the Levi-Civita of(M,h, i), then the Levi-Civita connection∇e forf^∗h, iis given by:

∇eXY :=f_∗⁻¹(∇f∗Xf_∗Y), (8) Whenf :M −→M is a homothetic transformation we get from its definition that h, iand f^∗h, ishare the same Levi-Civita connection i.e∇e =∇, which is to say thatany homothetic transformation is an affine transformation. Conversely:

Lemma 3.0.1. If (M,h , i) is an irreducible Riemannian manifold, then every affine transformationf :M −→M is homothetic.

Proof. According to (8), h , i and f^∗h , i define the same Levi-Civita ∇. Next recall that ifG⊂O(V,h, i)acts irreducibly on a Euclidean vector space(V,h, i) and preserves a symmetric bilinear form B : V ×V −→ R, then B =c²h , i for some constantc >0. Applying this result to

(V,h, i) = (TxM,h, ix), G= Hol_x(M,h, i) and B = (f^∗h, i)x, we obtain that for any x∈ M, (f^∗h , i)x =c²_xh, ix for some cx >0. Finally, if y∈M is another point andγ: [0,1]−→M is a piecewise smooth curve joiningx toy, then for everyv∈T_xM:

c²_yhτ^γ_x(v), τ^γ_x(v)iy = hTyf(τ^γ_x(v)), Tyf(τ^γ_x(v))if(y)

= hτ^f◦γ_f(x)(Txf(v)), τ^f◦γ_f(x)(Txf(v))i_f(y)

= hTxf(v), Txf(v)if(x)

= c²_xhv, vix.

Since hv, vix =hτ^γ_x(v), τ^γ_x(v)iy for any v ∈ TxM we get thatcx =cy, completing the proof.

Lemma 3.0.2. If(M,h, i)is a complete Riemannian manifold which is not locally Euclidean, then every homothetic transformation is an isometry.

Proof. Suppose that (M,h, i)admits a homothetic transformationf :M −→M that isn’t an isometry, and write f^∗h, i =c²h , i with c > 0. Next notice that f⁻¹ is homothetic as well with ration 1/c, therefore we suppose without loss of generality that0< c <1.

We start by proving that f has a fixed point. Denote d : M ×M −→ R⁺ the geodesic distance and take an arbitrary pointx∈M then put`:=d(x, f(x)). Let γ : [0,1]−→M be a minimizing geodesic joiningxto f(x), which exists since M is complete, thenfⁱ◦γ is a smooth curve joiningfⁱ(x)andfⁱ⁺¹(x)with length:

`i= Z 1

0

h(fⁱ◦γ)⁰(t),(fⁱ◦γ)⁰(t)i_f¹²i◦γ(t)dt=cⁱ`, Therefore ifm, n∈Nare such thatm < nthen:

d(f^m(x), fⁿ(x))≤

n−1

X

i=m

d(fⁱ(x), fⁱ⁺¹(x))≤

n+1

X

i=m

`i=

n+1

X

i=m

cⁱ`≤ c<sup>m</sup>` 1−c, and thus(f^m(x))mis a Cauchy sequence in(M, d)hence converges to somex^∗∈M sinceM is complete. Now x^∗ is obviously a fixed point off, furthermore x^∗ does not depend on the choice of x, indeed givenz ∈M andαa geodesic joining z to x^∗, we get that f^m◦αis a curve joiningf^m(z)tof^m(x^∗) =x^∗ and so:

d(f^m(z), x^∗)≤`(f<sup>m</sup>◦α) =c<sup>m</sup>`(α) −→

m→+∞0. (9)

Now fix a neighborhoodU ofx^∗ in M with compact closure. Then there exists a constantK^∗>0such that for anyy∈U and any unit vectorsv₁, v₂∈T_yM:

|hRy(v₁, v₂)v₁, v₂iy| ≤K^∗, (10)