Yamabe Constants of Collapsing Riemannian Submersions

Yamabe Constants of Collapsing Riemannian Submersions

Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.)

an der Fakult¨at f¨ur Mathematik der Universit¨at Regensburg

vorgelegt von Manuel Streil

aus Augsburg

Regensburg, im Juli 2016

Promotionsgesuch eingereicht am 6. Juli 2016

Die Arbeit wurde angeleitet von Prof. Dr. Bernd Ammann.

Pr¨ufungsausschuss:

Vorsitzender: Prof. Dr. Helmut Abels 1. Gutachter: Prof. Dr. Bernd Ammann

2. Gutachter: Prof. Dr. Mattias Dahl, KTH Stockholm weiterer Pr¨ufer: Prof. Dr. Stefan Friedl

Ersatzpr¨ufer: Prof. Dr. Clara L¨oh

3 Abstract

Letπ : (M^n+k, g)→(Bⁿ,g) be a surjective Riemannian submersion, whereˆ M and B are assumed to be closed, dimM ≥ 3, and the scalar curvature scal_g⊥ of every fibre F_b, b ∈ B, with respect to the induced metric g^⊥ is positive. We consider the metric r²gˆon B and rescale g on the horizontal subspaces accordingly to obtain a Riemannian submersion

π: (M, g_r2)→(B, r²g).ˆ

Then the limit of the Yamabe constants of (M, g_r²) exists and

r→∞lim Y(M,[g_r²]) = inf

b∈BY(Rⁿ×Fb,[geucl⊕g^⊥]).

If M is a smooth manifold of dimension dimM ≥3 that is the total space of a smooth fibre bundle with fibreF carrying a Riemannian metricgF such that scal_gF > 0 and structure group G = Isom(F, g_F), we obtain a lower bound for the Yamabe invariant ofM by

Y(M)≥Y(Rⁿ×F,[g_eucl⊕g_F]).

Zusammenfassung

Es sei π : (M^n+k, g) → (Bⁿ,g) eine surjektive Riemannsche Submersion,ˆ wobei wir annehmen, dass M and B kompakte Mannigfaltigkeiten ohne Rand sind und dimM ≥3.Außerdem sei die Skalarkr¨ummung scal_g⊥ jeder FaserF_b, b∈B,bez¨uglich der induzierten Metrikg^⊥positiv. Wir betrachten die Metrikr²ˆg aufB und erhalten nach entsprechender Reskalierung vong auf den Horizontalr¨aumen eine Riemannsche Submersion

π: (M, g_r²)→(B, r²g).ˆ

Dann existiert der Grenzwert der Yamabe-Konstanten von (M, g_r2),und es gilt

rlim→∞Y(M,[g_r2]) = inf

b∈BY(Rⁿ×F_b,[g_eucl⊕g^⊥]).

Ist nun M eine glatte Mannigfaltigkeit der Dimension dimM ≥ 3, die der Totalraum eines glatten Faserb¨undels ist, dessen Fasertyp F eine Rie- mannsche Metrik gF tr¨agt, sodass scalg_F > 0 und die Strukturgruppe G gleich Isom(F, g_F) ist, so erhalten wir eine untere Schranke f¨ur die Yamabe- Invariante vonM verm¨oge

Y(M)≥Y(Rⁿ×F,[g_eucl⊕g_F]).

Contents

1 Overview 7

1.1 The Yamabe Constant of a Conformal Manifold . . . 7

1.2 The Yamabe Invariant of a Manifold . . . 12

2 Riemannian Submersions 17 2.1 Preliminaries . . . 18

2.2 O’Neill’s Formulas for Curvature . . . 23

2.3 Rescaling the Metric . . . 37

2.4 Local Trivializations and Induced Metrics . . . 41

2.4.1 Lifting Properties . . . 41

2.4.2 Local Trivializations . . . 47

2.4.3 Induced Metrics and Estimates in Normal Coordinates 51 2.4.4 Admissible Trivializations . . . 57

2.4.5 Riemannian Submersions with Totally Geodesic Fibres 59 2.4.6 Fibre Bundles . . . 61

2.5 Integration . . . 63

3 Collapsing Riemannian Submersions 67 3.1 The Yamabe Constant . . . 67

3.2 Collapsing Riemannian Submersions . . . 69

5

Chapter 1

Overview

1.1 The Yamabe Constant of a Conformal Mani- fold

Let (Mⁿ, g) be a closed Riemannian manifold of dimension n ≥ 3. The famous Yamabe problem asks whether there exists a Riemannian metric ¯g conformal to g with constant scalar curvature.

This question was answered affirmatively by Aubin [Au], Schoen [Sch]

and Trudinger [Tr].

Writing ¯g=f^p−2·g withp=p_n= _n²ⁿ₋₂ andf ∈C^∞(M,R>0) and using the transformation rules for conformal changes one finds that

scal_¯g =s if and only if

Y^g(f) := ∆^gf+ n−2

4(n−1)·scal_g·f = n−2

4(n−1)·s·f^p−1, whereY^g is called conformal Laplacian.

It turns out that the nonlinear PDE Y^g(f) = λ·f^p−1 is the Euler- Lagrange equation of the Yamabe functional

Q(¯g) :=

�

Mscal_¯gdvol_g¯

��

Mdvolg¯�2/p, where ¯g varies in the conformal class [g].

Writing again ¯g=f^p−2 ·g for some function f ∈C^∞(M,R>0) and setting a=a_n= _4(nⁿ⁻₋²₁₎,one finds

Q(¯g) =Qg(f) :=

�

M

�₁

a�∇^gf�²g+ scal_g·f²� dvol_g

�f�²L^p(M,g)

.

7

We define theYamabe constantY(M,[g]) of [g] as Y(M,[g]) := inf

¯

g∈[g]Q(¯g) = inf{Q_g(f)|f ∈C^∞(M,R>0)}. Aubin showed (see Lemma 3.4 in [L-P]) that

Y(M,[g])≤Y(Sⁿ,[g_sph])

whereg_sph is the standard metric on the sphere Sⁿ⊂Rⁿ⁺¹. It turns out that

Y(M,[g]) = inf{Q_g(f)|f ∈C^∞(M)\ {0}}, which motivates the following

Definition 1.1. Let(Eⁿ, g) be a not necessarily compact Riemannian man- ifold without boundary of dimension n ≥ 3. Then we define its Yamabe constant as

Y(E,[g]) = inf{Qg(f)|f ∈C₀^∞(E)\ {0}}.

We note thatY(E,[g]) is indeed an invariant of the conformal class, since for allh∈C^∞(E,R^>0) we have

Q_g(hf) =Q_hp−2g(f).

Applying an approximation argument one can show that Y(Sⁿ,[g_sph]) = inf�

Q_gsph(f)|f ∈ F� where

F :=�

f ∈C^∞(Sⁿ)\ {0} |f|_Bρ(q)= 0 for a ρ >0� withq∈Sⁿ fixed.

Using stereographic projection one deduces that Y(Sⁿ,[g_sph]) = Y(Rⁿ,[g_eucl])

= 1

a·inf

��∇ϕ�²_L2

�ϕ�²_L^p

��

��ϕ∈C₀^∞(Rⁿ)

� .

In other words, computingY(Sⁿ,[g_sph]) is equivalent to finding the best constant in the Sobolev inequality, which is realized by a family of spherically symmetric functions (see e.g. the appendix to chapter V in [S-Y2]).

It follows that

Y(Sⁿ,[g_sph]) =n(n−1)vol(Sⁿ)^2/n.

Moreover, using Obata’s lemma we have that the Yamabe functional on (Sⁿ, g_sph) is minimized by constant multiples of the standard metric and its

1.1. THE YAMABE CONSTANT OF A CONFORMAL MANIFOLD 9 images under conformal diffeomorphisms. These are the only metrics con- formal to the standard on onSⁿ that have constant scalar curvature.

We remark that Aubin’s lemma above carries over to noncompact man- ifolds (Mⁿ, g) of dimension n≥3 without boundary.

Considering Riemannian products, Akutagawa, Florit and Petean proved in [A-F-P] the following

Theorem 1.2. Let (M^m, g) be a closed Riemannian manifold of dimension m ≥ 2 with positive scalar curvature scalg > 0 and (Nⁿ, h) any closed Riemannian manifold. Then

rlim→∞Y(M ×N,[g⊕r²h]) =Y(M×Rⁿ,[g⊕g_eucl]).

We briefly sketch the proof of Theorem 1.2. Due to the compactness of M one obtains an r₀ and a constant c > 0 such that scal_g⊕r2h > c for all r > r0 and hence

Y(M×N,[g⊕r²h])>0 for all r0 >0

by Remark 3.1. Moreover, (M ×Rⁿ, g⊕g_eucl) being a complete Rieman- nian manifold with strictly positive injectivity radius and bounded sectional curvature, by Theorem 2.21 in [Au] there is a continuous embedding

W^1,2(M×Rⁿ, g⊕g_eucl)�→L^p(M×Rⁿ, g⊕g_eucl), which then yields the key observation

Y(M×Rⁿ,[g⊕g_eucl])>0.

Now one considers normal coordinates with respect to the rescaled metric r²honN and uses a linear isometry to identify ballsB_rεⁿ(0)⊂Rⁿwith balls B_ε^h(0) = B^r_rε^2h(0) = V ⊂TqN, where q ∈N and ε >0 is sufficiently small such that uniform estimates inr can be made between the euclidean metric on B_rεⁿ(0) andr²h in normal coordinates on

exp^h_q(V) = exp^r_q^2h(V).

Givenδ >0 there exists anε >0 such that (1 +δ)^−n/2 <

�

det((r²h)_ij(x))<(1 +δ)^n/2, 1

1 +δ�η�² <

�n ij=1

(r²h)_ijη_iη_j <(1 +δ)�η�², 1

1 +δ�η�² <

�n ij=1

(r²h)^ijη_iη_j <(1 +δ)�η�²

for allx∈B_rεⁿ(0) andη= (η1, . . . , ηn)∈Rⁿ.

This allows to compare test functions in C₀^∞(M ×Rⁿ) with test functions inC^∞(M ×N).

One proceeds by showing lim sup

r→∞ Y(M×N,[g⊕r²h])≤Y(M ×Rⁿ,[g⊕g_eucl]) and

Y(M×Rⁿ,[g⊕g_eucl])≤lim inf

r→∞ Y(M×N,[g⊕r²h]), which yields the theorem.

We significantly generalize the theorem above and replace Riemannian products by surjective Riemannian submersions such that all fibres have positive scalar curvature with respect to the induced metric.

Given a surjective Riemannian submersion (M, g)→(B,ˆg),

where dimM ≥3,we have for anyp∈M ag−orthogonal decomposition T_pM =T_pF_b⊕ Hp,

whereb=π(p), F_b =π⁻¹(b).

Using that dπ_p : Hp → T_bB is a linear isometry we may replace for r > 0 the metricg on the horizontal subspaces Hp by the pullback of r²ˆg under dπto obtain a metric g_r² on M such that

π: (M, g_r2)→(B, r²g)ˆ is a Riemannian submersion.

We investigate Y(M,[g_r2]) for r → ∞ while assuming that all fibres of π:M →B have positive scalar curvature with respect to the metric induced byg. It turns out that the limit exists. More precisely, we have

Theorem 1.3. Let π : (M^n+k, g) → (Bⁿ,g)ˆ be a surjective Riemannian submersion, where M and B are assumed to be closed, dimM ≥3,and the scalar curvature scal_g⊥ of every fibre F_b, b∈B, with respect to the induced metricg^⊥ is positive. Considering the Riemannian submersion

π: (M, g_r2)→(B, r²g)ˆ we have

rlim→∞Y(M,[g_r2]) = inf

b∈BY(Rⁿ×F_b,[g_eucl⊕g^⊥]).

1.1. THE YAMABE CONSTANT OF A CONFORMAL MANIFOLD 11 The proof of the theorem above is much more involved than the product case, but follows a similar pattern taking into account that all the arguments therein are local in nature.

We prove the inequalities lim sup

r→∞ Y(M,[g_r2])≤ inf

b∈BY(Rⁿ×F_b,[g_eucl⊕g^⊥]) and

binf∈BY(Rⁿ×F_b,[g_eucl⊕g^⊥)]≤lim inf

r→∞ Y(M,[g_r2]) in Proposition 3.4 and Proposition 3.5, respectively.

As we will see in Corollary 2.18, there exists an r₀>0 such that scal_gr2 >0 for all r > r₀,

which yields

Y(M,[g_r2])>0.

As above in the product case we have

Y(Rⁿ×F_b,[g_eucl⊕g^⊥])>0 for all b∈B.

Following Herrmann (Theorem 9.42 in [Be]) we use that horizontal lifts of geodesics inB are geodesics inM, choose ε >0 such that

exp^ˆg_b = exp^r_b^2ˆg =: exp_b:TbB ⊃U :=B_ε^ˆg(0) =B_rε^r2gˆ(0)→exp_b(U) =:V is a diffeomorphism and construct in section 2.4.2 a local trivialization

Ψ :V ×F_b→π⁻¹(V) in a neighbourhood ofb∈B by lifting geodesics.

Using Ψ andr²gˆ−normal coordinates centered atb∈V we identify test functions on Rⁿ×F_b with test functions onπ⁻¹(V) for sufficiently larger.

Vice versa, given a test function f on M, we find due to the compactness ofB finitely manyb_i and associated trivializations Ψ_i :V_i×F_bi →π⁻¹(V_i).

After choosing a partition of unity{ηi} subordinated to{Vi} we are able to identifyη_i·f with a test function onRⁿ×F_bi.

In order to prove the claimed inequalities we have to compare the Yam- abe functionals onV ×F_b with respect to the product metricr²gˆ⊕g^⊥and the induced metric Ψ^∗g_r2,respectively.

As a key observation we recognize in Lemma 2.29 that r²gˆ⊕g^⊥���

(b,p) = Ψ^∗g_r2|(b,p)

for any r >0 and p∈F_b.

Now we choose ε > 0 sufficiently small and normal coordinates near p on F_b and near b on B to obtain estimates for the local representation of the metricsr²gˆ⊕g^⊥ and Ψ^∗g_r².

This allows us in section 2.4.4 to compare the volume elements and the gradients of test functions on such ”admissible” trivializationsV ×F_b with respect to the metricsr²gˆ⊕g^⊥ and Ψ^∗g_r2.

1.2 The Yamabe Invariant of a Manifold

Definition 1.4. We define the Yamabe invariant of a smooth manifold M of dimM ≥3 as

Y(M) := sup

g Y(M,[g]),

where the supremum is taken over all Riemannian metrics g on M.

One reason why one is interested in the Yamabe invariant is that a smooth manifold of dimensionn≥3 admits a metric of positive scalar cur- vature if and only ifY(M)>0.

Due to Aubin one has an upper bound

Y(M)≤Y(Sⁿ) =Y(Sⁿ,[g_sph]).

In dimensionn≥5 it is an open question whether there is a closed manifold satisfying Y(M)�= 0 and Y(M)�=Y(Sⁿ), but one expects that many such manifolds exist.

Concerning lower bounds for the Yamabe invariant, Ammann, Dahl and Humbert showed

Theorem 1.5 (Corollary 1.4 in [A-D-H1]). If N is obtained from a closed n−dimensional manifold M by k−dimensional surgery,k≤n−3,then

Y(N)≥min{Λ_n,k, Y(M)},

where Λ_n,k is a positive number that depends only on n andk. In addition, Λ_n,0 =Y(Sⁿ).

This theorem generalizes previous results by Gromov-Lawson [G-L] and Schoen-Yau [S-Y1], Kobayashi [Ko], Petean [Pe1] and Petean-Yun [P-Y].

1.2. THE YAMABE INVARIANT OF A MANIFOLD 13 It allows several applications by using methods from bordism theory, see section 1.4 in [A-D-H1]. For these applications it is important to have ex- plicit lower bounds for the Yamabe invariant of HP²-bundles. Having this in mind we study the following situation:

Suppose we have a smooth fibre bundle π : M → B whith fibre F carrying a Riemannian metric gF and structure group G = Isom(F, gF).

Given a metric ˆgonB we apply Lemma 2.35 and find a Riemannian metric g onM such that

π: (M, g)→(B,g)ˆ

is aRiemannian submersionwith all fibres (F_b, g_b^⊥) beingisometricto (F, g_F).

By Theorem 1.3 we obtain

Corollary 1.6. Let M be a smooth manifold of dimension dimM ≥ 3 and suppose thatM is the total space of a smooth fibre bundle with fibre F carrying a Riemannian metricg_F such that scal_gF >0 and structure group Gequal toIsom(F, g_F).Then a lower bound for the Yamabe invariant ofM is given by

Y(M)≥Y(Rⁿ×F,[g_eucl⊕g_F]) where n is the dimension of the base space.

As mentioned above this corollary is particularly interesting if F is the quaternionic projective planeHP² equipped with its standard metricg_HP2. Note that PSp(3) acts by isometries onHP².Stolz proved

Theorem 1.7 (Theorem B in [St]). Let M be a compact spin manifold of dimension n ≥ 5. We assume that the index α(M) ∈ KO_n(pt) vanishes.

Then M is spin-bordant to the total space of an HP²-bundle over a base B such that the structure group is PSp(3).

In the following we assume that n ≥ 11. Let M0 be the total space of a bundle with fibre HP² and structure group PSp(3) over a base B of dimensionn−8.Applying Corollary 1.6 yields

Y(M0)≥Y(HP²×Rⁿ⁻⁸,[g_HP² ⊕g_eucl]).

Ammann, Dahl, Humbert [Theorem 2.3 in [A-D-H2]] estimated Y(HP²×Rⁿ⁻⁸,[g_HP2⊕g_eucl])≥

≥ n/an

(8/a8)^8/n((n−8)/an−8)^(n−8)/n ·Y(HP²,[g_HP²])^8/n·Y(Sⁿ⁻⁸,[g_sph])^(n−8)/n. Using Corollary 1.6 they [Proposition 6.8 in [A-D-H3]] were able to prove that

Y(M₀)≥ n a_n

�3⁶·2¹⁸ 7⁸·5² ·π⁸

�1/n

ν_n^1/n₋₈,

wherea_n= _4(n−1)ⁿ⁻² and ν_j =�_Y(Sj,[gsph]) j/aj

�j

.

Now suppose thatM is a compact, simply connected spin manifold with α(M) = 0.ThenM is spin-bordant to the total spaceM₀of anHP²-bundle with structure group PSp(3) by Theorem 1.7. Moreover,M can be obtained from M₀ by performing a sequence of surgeries of dimensions 0, . . . , n−3.

Consequently, we can estimate

Y(M) ≥ min{Λn,1, . . . ,Λn,n−3, Y(M0)}

≥ min�

Λ_n,1, . . . ,Λ_n,n−3, Y(HP²×Rⁿ⁻⁸,[g_HP2⊕g_eucl])� . (Compare also Proposition 6.9 in [A-D-H3].)

This application is currently the most important one of this PhD thesis.

For sake of completeness we also comment on dimensionn≤10.

Consideringn∈ {9,10} we first note

Lemma 1.8 (Lemma 5.5 in [A-D-H4]). Let M be a compact 2-connected spin manifold of dimension n ∈ {9,10}, which has α(M) = 0. Then M is obtained fromS⁹ oder HP²×S¹ (for n=9) or from S¹⁰ or HP²×S¹×S¹ (for n=10) by a sequence of surgeries of dimensions k ∈ {0,1, . . . , n−4}. All these surgeries are compatible with orientation and spin structure.

Lets₁ :=Y(HP²×S¹) ands₂ :=Y(HP²×S¹×S¹).

Ammann, Dahl und Humbert showed

Corollary 1.9 (Corollary 5.6 in [A-D-H4]). Let M be a 2−connected com- pact spin manifold of dimensionn= 9 or n= 10 withα(M) = 0. Then

Y(Mⁿ⁼⁹)≥min{Λ_9,1,Λ_9,2,Λ_9,3,Λ_9,4,Λ_9,5, s₁}>109.2 and

Y(Mⁿ⁼¹⁰)≥ {Λ_10,1,Λ_10,2,Λ_10,3,Λ_10,4,Λ_10,5,Λ_10,6, s₂} ≥97.3.

By [Theorem 1.2 in [Pe2]] we have

s1 ≥Y(HP²×R,[g_HP² ⊕g_eucl])≥0.9370·Y(S⁹,[g_sph]) = 138.57... >109.2, and using [Example after Theorem 1.7 in [P-R]] it follows

s₂ ≥Y(HP²×R²,[g_HP2 ⊕g_eucl])≥0.59·Y(S¹⁰,[g_sph])>97.3<Λ_10,1. In dimension n = 8 the only HP²-bundles in the sense of Stolz are compact manifolds the connected components of which are diffeomorphic to

1.2. THE YAMABE INVARIANT OF A MANIFOLD 15 HP². Stolz’ HP²-bundles have to carry an orientation, and the diffeomor- phism may either be orientation preserving or orientation reversing. There- fore we have

Y(M)≥Y(HP²,[g_HP²]).

We note thatY(HP²,[g_HP²]) can be computed explicitly, since (HP², g_HP²) is an Einstein manifold and Obata’s lemma applies.

In dimensionn∈ {5,6,7}the total space of theHP²-bundle in Theorem 1.7 is the empty set, thus the phenomena discussed in our PhD thesis do not play a major role in this case.

Acknowledgements: While writing this PhD thesis the author was supported by the Graduiertenkolleg Curvature, Cycles and Cohomology.

The author would like to thank his adviser Prof. Dr. Bernd Ammann for suggesting the interesting topic, inspiring discussions and support through- out the years.

Chapter 2

Riemannian Submersions

We will give a self-contained introduction with complete proofs to Rieman- nian submersions

π : (M, g)→(B,g)ˆ , which are always assumed to besurjective.

The main reference is chapter 9, sections A-F in [Be] and chapter II, section 6 in [Sa].

After dealing with some very basic concepts concerning horizontal and ver- tical vector fields we introduce in section 2.1 the tensorsAand T which are obstructions to the horizontal distribution to be integrable and the fibres to be totally geodesic, respectively.

In section 2.2 we prove O’Neill’s formulas for curvature, which relate the curvature tensors ofM, B and the fibres, and compute afterwards the sec- tional, Ricci and scalar curvature of M. We discuss in section 2.3 how the curvatures change if we rescale the metric onBand the horizontal subspaces accordingly to obtain a Riemannian submersion

π: (M, g_r²)→�

B, r²gˆ� .

We remark that in contrast to [Be] we vary the metric on the horizontal subspaces and not on the vertical subspaces. The formula for scal_gr2 in Proposition 2.17 shows that for large r the scalar curvature of the fibres dominates, which yields in Corollary 2.18 a metric of positive scalar curva- ture onM provided thatM andB are compact and scal_g⊥ >0 for all fibres.

In section 2.4 we investigate lifting properties of curves and prove that geodesics on B have unique lifts to horizontal geodesics on M if B and M are compact, which yields a local product structure. After adjusting the trivialization neighbourhood V near b∈B we are able to compare the product metric r²gˆ⊕g^⊥ with the metric induced by g on V ×F_b, which will be crucial for the estimates in chapter 3. Afterwards we explain how to construct Riemannian submersions with isometric fibres from fibre bundles having fibre (F, gF) and structure group Isom(F, gF).

17

We end the chapter by proving a generalization of Fubini’s theorem for Rie- mannian submersions. This section is based on section II.5 in [Sa].

2.1 Preliminaries

Let

π: (M, g)→(B,ˆg)

be a submersion between Riemannian manifoldsM^n+k, Bⁿwith Levi-Civita connections ∇ and ˆ∇. Due to the implicit function theorem we find for any p ∈ M an ε > 0 and charts ϕ : p ∈ U → ϕ(U) = (−ε, ε)^n+k and ψ:b=π(p)∈V →ψ(V) = (−ε, ε)ⁿ such thatπ(U)⊂V and

ψ◦π◦ϕ⁻¹(x₁, . . . , x_n, x_n+1, . . . , x_n+k)�→(x₁, . . . , x_n).

We note that every fibreF_b =π⁻¹(b)⊂M is ak−dimensional submanifold of M with induced metricg^⊥.A chart is given by the composition

ϕ_b :F_b∩U −→^ϕ (−ε, ε)^n+k→(−ε, ε)^k q�→(ψ(b), y₁, . . . , y_k)�→(y₁, . . . , y_k).

The tangent subspace to F_b in T_pM is the vertical subspace Vp = T_pF_b at p,whereas the horizontal subspaceatpis the orthogonal complementHp to Vp inT_pM,the elements of which are called verticaland horizontal vectors, respectively. Given v ∈T_pM we have a unique decompostion v =v^�+v^⊥ with v^� ∈ Hp and v^⊥ ∈ Vp. As the union of these spaces we obtain the vertical distributionV and thehorizontal distribution H.

In the following π :M →B will be a Riemannian submersion, i. e. the induced isomorphism

dπ_p :Hp→T_π(p)M

is aRiemannian isometryfor every p∈M, so that the length of horizontal vectors is preserved. It follows that every vector field ˆX onB has a unique smooth horizontal liftX.

We call a vector field X onM basicif there exists a vector field ˆX onB such that dπ_pX_p = ˆX_π(p) for every p∈M. In other words the vector fields X and ˆX areπ−related.

We make the following elementary observations:

1. Let X and Y be basic vector fields which induce ˆX = dπ(X) and Yˆ = dπ(Y). Then we have [ ˆX,Yˆ] = dπ[X, Y] = dπ([X, Y]^�), and [X, Y]^� is the horizontal lift of [ ˆX,Yˆ].

2. For a basicX and vertical U we obtain dπ([X, U]^�) = 0,so [X, U] is vertical.

2.1. PRELIMINARIES 19 3. If X and Y are basic and U is vertical, then U g(X, Y) = 0, because

the inner product is constant along the fibres.

Whereas the vertical distribution V is integrable in the sense of Frobe- nius, the horizontal distributionH need not be. In fact, we have

Lemma 2.1 (Proposition 9.24 in [Be]). For any horizontal vector fields X and Y the equality

(∇xY)^⊥= 1

2[X, Y]^⊥ holds.

Proof. We observe that the expressions (∇xY)^⊥ and ¹₂[X, Y]^⊥ are tensorial in X and Y, so we may assume X and Y to be basic. For any vertical U Koszul’s formula yields

2g�

(∇XY)^⊥, U�

= 2g(∇XY, U)

= Xg(Y, U) +Y g(U, X)−U g(X, Y)

+g([X, Y], U)−g([Y, U], X) +g([U, X], Y)

= g�

[X, Y]^⊥, U� and the formula is proven.

Following O’Neill we embed (X, Y) �→ (∇xY)^⊥ into a tensor field A of type (2,1) on M.

Definition 2.2 ((9.20) in [Be]). For vector fieldsE and F onM we set A_EF :=�

∇E^�F^⊥�_� +�

∇E^�F^��_⊥ .

As a basic observation we remark that A_X is alternating, since g(A_XE, F) = g�

∇XE^⊥, F^�� +g�

∇XE^�, F^⊥�

= −g�

E^⊥,(∇XF^�)^⊥�

−g�

E^�,(∇XF^⊥)^��

= −g�

E,(∇XF^�)^⊥�

−g�

E,(∇XF^⊥)^��

= −g(E, A_XF). Furthermore

A_XY = 1

2[X, Y]^⊥ =−1

2[Y, X]^⊥=−A_YX.

Lemma 2.3. The horizontal distributionHis integrable if and only ifA≡0.

Proof. If A ≡ 0, then [X, Y]^⊥ = 2A_XY = 0, and [X, Y] is horizontal.

AssumingHto be integrable we obtain 0 =g

�1

2[X, Y]^⊥, U

�

=g(A_XY, U) =−g(Y, A_XU)

for any horizontal X, Y and vertical U. Consequently, A_XU = 0, and it followsA≡0.

The Levi-Civita connection∇on (M, g) induces the Levi-Civita connec- tion∇^⊥ on each fibre given by

∇^⊥UV = (∇UV)^⊥.

We embed the horizontal part (∇UV)^�,i. e. the second fundamental form of the fibres, into a tensor fieldT of type (2,1) onM.

Definition 2.4 ((9.17) in [Be]). For vector fieldsE and F onM we set T_EF :=�

∇E^⊥F^⊥�_� +�

∇E^⊥F^��_⊥ . We remark that T_U is alternating, as

g(T_UE, F) = g�

∇UE^⊥, F^�� +g�

∇UE^�, F^⊥�

= −g�

E^⊥,(∇UF^�)^⊥�

−g�

E^�,(∇UF^⊥)^��

= −g�

E,(∇UF^�)^⊥�

−g�

E,(∇UF^⊥)^��

= −g(E, T_UF). Moreover

TUV −TVU = (∇UV − ∇VU)^�= [U, V]^�= 0.

It follows

T_UV =T_VU.

SinceT_UV is the second fundamental form of the fibres, we obtain Lemma 2.5. Each fibre is totally geodesic if and only ifT ≡0.

Proof. Firstly, we assume that T vanishes identically. Let γ : (−ε, ε)→ F_b be a geodesic in some fibre (F_b, g^⊥,∇^⊥) andt0 ∈(−ε, ε).Using Lemma 2.22 below and the fact that a Riemannian submersion is locally a projection we obtain a vertical extensionU of γ^�(t₀) ∈ Vγ(t0) in a neighbourhood U ⊂M such that∇^⊥_UU = 0 inU ∩F_b.An integral curve

η: (−ε, ε)⊃(t0−ε^�, t0+ε^�)→ U

2.1. PRELIMINARIES 21 of U withη(t0) =γ(t0) is then a geodesic in (F_b, g^⊥,∇^⊥) and

η(t) =γ(t) for all t∈(t₀−ε^�, t₀+ε^�).

Furthermore,T ≡0 implies∇UU =∇^⊥_UU+TUU = 0 inU∩Fb.Consequently, η and hence γ is a geodesic in (M, g,∇) through γ(t₀). In other words, (F_b, g^⊥) is a totally geodesic submanifold of (M, g).

Conversely, let U, V be vertical andX horizontal. Then g(T_UX, V) =−g(X, T_UV)

and

T_U+V(U +V) =T_UU+T_VV +T_UV +T_VU =T_UU +T_VV + 2·T_UV.

Therefore, it suffices to show T_UU = 0 for all vertical vector fields U if the fibres are totally geodesic. Let p∈ M and Up ∈ Vp. Applying Lemma 2.22 as above we find a vertical extension U of U_p in a neighbourhood U ⊂M such that ∇^⊥_UU = 0 in U ∩F_π(p). An integral curve γ : (−ε, ε) → U of U withγ(0) =pis then a geodesic of (M, g) and takes values in the fibreF_π(p), sinceγ^�(0) =U_p.It follows (∇UU)_p = 0 and

(TUU)_p= (∇UU)_p−�

∇^⊥UU�

p = 0.

Lemma 2.6 (9.32 in [Be]). For an arbitrary vector field E on M, vertical U, V and horizontal X, Y we have

g((∇EA) (X, Y), U) = −g((∇EA) (Y, X), U) g((∇ET) (U, V), X) = g((∇ET) (V, U), X).

Proof. We use A_XY = −A_YX, which implies ∇E(A_XY) = −∇E(A_YX) and

A_∇EXY =−A_Y (∇EX)^�=−A_Y (∇EX) +�

∇Y (∇EX)^⊥�_� . So we obtain

g(A_∇EXY, U) =−g(AY(∇EX), U) and

g(A_∇EYX, U) =−g(A_X(∇EY), U), respectively. Similarly,

TUV =TVU yields ∇E(TUV) =∇E(TVU)

and

T_∇EUV =T_V (∇EU)^⊥=T_V (∇EU)− ∇V

�

(∇EU)^��_⊥ . As a result,

g(T_∇EUV, X) =g(T_V (∇EU), X) and g(T_∇EVU, X) =g(T_U(∇EV), X), respectively.

We conclude the section with

Lemma 2.7 ((6.5) in [Sa]). Let Xˆ and Yˆ be vector fields on B with hori- zontal lifts X andY on M. Then (∇XY)^� is the horizontal lift of ∇ˆXˆY .ˆ Proof. By Koszul’s formula we have

2ˆg�

∇ˆXˆY ,ˆ Zˆ�

= Xˆˆg� Y ,ˆ Zˆ�

+ ˆYgˆ� Z,ˆ Xˆ�

−Zˆˆg� X,ˆ Yˆ�

= +ˆg��

X,ˆ Yˆ� ,Zˆ�

−gˆ��

Y ,ˆ Zˆ� ,Xˆ�

+ ˆg��

Z,ˆ Xˆ� ,Yˆ�

. Since dπ_p : Hp → T_π(p)M is a Riemannian isometry for every p ∈ M, we obtain

X_pg(Y, Z) = ˆX_π(p)ˆg� Y ,ˆ Zˆ�

,

Y_pg(Z, X) = ˆY_π(p)gˆ� Z,ˆ Xˆ� and

Z_pg(X, Y) = ˆZ_π(p)gˆ� X,ˆ Yˆ�

. Moreover,�

X,ˆ Yˆ�

=dπ�

[X, Y]^��

implies ˆ

g_π(p)��

X,ˆ Yˆ� ,Zˆ�

=g_p([X, Y], Z). Analogously,

ˆ g_π(p)��

Y ,ˆ Zˆ� ,Xˆ�

=g_p([Y, Z], X) and

ˆ g_π(p)��

Z,ˆ Xˆ� ,Yˆ�

=g_p([Z, X], Y). It follows ˆg_π(p)�

∇ˆXˆY ,ˆ Zˆ�

=g_p�

(∇XY)^�, Z�

= ˆg_π(p)�

dπ_p(∇XY)^�_p ,Zˆ_π(p)� and consequently�

∇ˆXˆYˆ�

π(p)=dπ_p(∇XY)^�_p .

2.2. O’NEILL’S FORMULAS FOR CURVATURE 23

2.2 O’Neill’s Formulas for Curvature

LetR,RˆandR^⊥the curvature tensors corresponding to (M, g,∇),�

B,ˆg,∇ˆ� and the fibres�

F_b, g^⊥,∇^⊥�

,respectively. We give complete proofs for O’Neill’s formula for curvature (cf. Proposition 6.2 in [Sa])

Formula 1. Let U, V, W andW^� be vertical. Then g�

R(U, V)W, W^��

= g�

R^⊥(U, V)W, W^�� +g�

T_UW, T_VW^��

−g�

T_VW, T_UW^�� Proof. By definition we have

R^⊥(U, V)W =∇^⊥U

�∇^⊥VW�

− ∇^⊥V

�∇^⊥UW�

− ∇^⊥_[U,V]W.

Since U, V and W are vertical, we obtain ∇^⊥U

�∇^⊥VW�

=�

∇U(∇VW)^⊥�_⊥ ,

∇^⊥_V �

∇^⊥_UW�

=�

∇V (∇UW)^⊥�_⊥

and ∇^⊥_[U,V]W =�

∇[U,V]W�_⊥ . It follows

g�

R^⊥(U, V)W, W^��

= g�

∇U(∇VW)^⊥− ∇V (∇UW)^⊥− ∇[U,V]W, W^��

= g�

R(U, V)W, W^��

−g�

∇U(∇VW)^�, W^�� +g(∇V(∇UW)^�, W^�)

= g�

R(U, V)W, W^�� +g�

(∇VW)^�,�

∇UW^��_��

−g�

(∇UW)^�,�

∇VW^��_��

= g(R(U, V)W, X) +g�

T_VW, T_UW^��

−g�

T_UW, T_VW^�� .

Formula 2. Let U, V, W be vertical and X be horizontal. Then g(R(U, V)W, X) =g((∇UT) (V, W), X)−g((∇VT) (U, W), X) Proof. With

(∇UT) (V, W) =∇U(T_VW)−T_∇UVW −T_V (∇UW) and

(∇VT) (U, W) =∇V (TUW)−T_∇VUW −TU(∇VW)

we get (∇UT) (V, W)−(∇VT) (U, W) =

= ∇U(∇VW)^�− ∇V (∇UW)^�−T_[U,V]W

−�

∇V (∇UW)^⊥�_�

−�

∇V (∇UW)^��_⊥ +�

∇U(∇VW)^⊥�_�

−�

∇U(∇VW)^��_⊥ . Hence,g((∇UT) (V, W)−(∇VT) (U, W), X) =

= g�

∇U(∇VW)^�, X�

−g�

∇V (∇UW)^�, X�

−g�

∇[U,V]W, X�

−g�

∇V (∇UW)^⊥, X� +g�

∇U(∇VW)^⊥, X�

= g�

∇U∇VW − ∇V∇UW − ∇[U,V], X�

= g(R(U, V)W, X).

Formula 3. Let U, V be vertical and X, Y be horizontal. Then g(R(U, X)Y, V) = g((∇XT) (U, V), Y)−g(T_UX, T_VY)

+g((∇UA) (X, Y), V) +g(A_XU, A_YV). Proof. Since we are dealing with tensors, we may assume X and Y to be basic vector fields.

We calculateg((∇XT) (U, V), Y) =

= g�

∇X(∇UV)^�, Y�

−g(T_∇XUV, Y)−g�

∇U(∇XV)^⊥, Y�

= g(R(X, U)V, Y)−g�

∇X(∇UV)^⊥, Y�

−g(T_∇XUV, Y) +g(∇U(A_XV), Y) +g�

∇[X,U]V, Y�

Using [X, U]^�= 0 and [X, U] =∇XU− ∇UX we obtain g�

∇[X,U]V, Y�

=g(T_∇XUV, Y)−g(T_∇UXV, Y) and consequently

g((∇XT) (U, V), Y) = g(R(X, U)V, Y)−g�

∇X(∇UV)^⊥, Y� +g(∇U(A_XV), Y)−g(T_∇UXV, Y). Furthermore,

g(T_∇UXV, Y) =g�

T_V (∇UX)^⊥, Y�

=−g(T_VY, T_UX)

2.2. O’NEILL’S FORMULAS FOR CURVATURE 25 and

g�

∇X(∇UV)^⊥, Y�

= −g(∇UV, A_XY)

= −U g(V, A_XY) +g(V,∇U(A_XY)) together with

g(∇U(A_XV), Y) = U g(A_XV, Y)−g(A_XV,∇U Y)

= −U g(A_XY, V) +g(A_X(∇UY), V) imply

g(R(X, U)V, Y) = g((∇XT) (U, V), Y)−g(T_VY, T_UX) +g(V,∇U(A_XY))−g(A_X(∇UY), V)

= g((∇XT) (U, V), Y)−g(T_VY, T_UX) +g((∇UA) (X, Y), V) +g(A_∇UXY, V). Finally,

g(A_∇UXY, V) = g� A_(∇

UX)^�Y, V�

=g� A_(∇

XU)^�Y, V�

= −g�

AY (∇XU)^�, V�

=g(AYV, AXU) yields

g(R(X, U)V, Y) = g(R(U, X)Y, V)

= g((∇XT) (U, V), Y)−g(T_UX, T_VY) +g((∇UA) (X, Y), V) +g(A_XU, A_YV).

Formula 4. Let U, V be vertical and X, Y horizontal. Then

g(R(U, V)X, Y) = g((∇VA) (X, Y), U)−g((∇UA) (X, Y), V) +g(A_XV, A_YU)−g(A_XU, A_YV)

+g(TUX, TVY)−g(TVX, TUY) and

Proof. We may assumeX and Y to be basic, so [V, X]^�= [U, Y]^�= 0 and consequently

g� A_(∇

VX)^�Y, U�

= g� A_(∇

XV)^�Y, U�

=−g�

A_Y (∇XV)^�, U�

= g(AYU, AXV).

It followsg((∇VA) (X, Y), U) =

= g(∇V (A_XY)−A_X(∇VY), U)−g(A_YU, A_XV)

= g(R(V, X)Y, U)−g�

∇V (∇XY)^�, U� +g�

∇X(∇VY)^⊥, U�

−g(AYU, AXV) +g�

∇[V,X]Y, U�

andg((∇VA) (X, Y), U) =g((∇UA) (X, Y), V) =

= g(R(V, X)Y, U) +g(R(Y, V)X, U)−g�

∇V (∇XY)^�, U�

−g(A_YU, A_XV) +g�

∇X(∇VY)^⊥, U�

+g(A_YV, A_XU) +g�

∇U(∇XY)^�, V�

−g�

∇X(∇UY)^⊥, V� +g�

∇[V,X]Y, U�

−g�

∇[U,X]Y, V� . Moreover,g�

∇X(∇VY)^⊥, U�

=

= Xg(T_VY, U)−g�

T_VY,(∇XU)^��

= g(T_VY, U)−g(T_VY, T_UX) +g�

T_VY,[U, X]^⊥�

= Xg(T_VY, U)−g(T_VY, T_UX) +g�

∇[U,X]Y, V� , where we used

g�

T_VY,[U, X]^⊥�

= −g�

T_V [U, X]^⊥, Y�

=−g�

T_[U,X]⊥V, Y�

= g�

T_[U,X]⊥Y, V�

=g�

∇[U,X]Y, V� .

Analogously, g�

∇X(∇UY)^⊥, V�

=Xg(T_UY, V)−g(T_UY, T_VX) +g�

∇[V,X]Y, U� . We calculate

Xg(T_VY, U)−Xg(T_UY, V) =Xg(T_UV −T_VU, Y) = 0 and

g�

∇U(∇XY)^�, V�

= g�

T_U(∇XY)^�, V�

=−g�

T_UV,(∇XY)^��

= −g�

T_VU,(∇XY)^��

=g�

T_V (∇XY)^�, U�

= g�

∇V (∇XY)^�, U� .

2.2. O’NEILL’S FORMULAS FOR CURVATURE 27 Finally, we use Bianchi’s identity

g(R(V, X)Y, U) +g(R(Y, V)X, U) = −g(R(X, Y)V, U)

= g(R(U, V)X, Y) and obtain the claimed formula.

Formula 5. Let X, Y, Z be horizontal and U vertical. Then

g(R(X, Y)Z, U) = −g((∇ZA) (X, Y), U)−g(TUZ, AXY) +g(T_UY, A_ZX) +g(T_UX, A_YZ)

Proof. Letp∈M. We apply Corollary 2.21 below and choose extensions of dπ_pX_p = ˆX_π(p) and dπ_pY_p = ˆY_π(p) to local vector fields ˆX and ˆY on B such

that �

∇ˆXˆYˆ�

π(p)=�

∇ˆYˆXˆ�

π(p)= 0.

We may assume that X and Y are the horizontal lifts of ˆX and ˆY near p. Since (∇XY)^� is the horizontal lift of ˆ∇XˆY ,ˆ it follows (∇XY)^�_p = 0. In other words, we can choose local extensions X, Y and Z of X_p, Y_p and Z_p such that

(∇XY)^�_p = (∇YX)^�_p = (∇XZ)^�_p = (∇ZX)^�_p = (∇ZY)^�_p = (∇YZ)^�_p = 0.

It follows

g_p(A_Z(∇XY), U) =g_p�

∇Z(∇XY)^�, U�

=−g_p�

(∇XY)^�,∇ZU�

= 0.

Combining with (A_∇ZXY)_p =A_(∇ZX)�pY_p= 0 and g_p(A_X(∇ZY), U) =g_p�

A_Xp(∇ZY)^�_p , U_p�

= 0 we have

g_p((∇ZA) (X, Y), U) =g_p(∇Z(A_XY), U). Using (A_YZ)_p = ¹₂�

(A_YZ)_p−(A_ZY)_p�

= ¹₂�

(∇YZ)_p−(∇ZY)_p�

we get gp(∇X(AYZ), U) = Xpg(AYZ, U)−gp(AYZ,∇XU)

= 1

2g(∇YZ− ∇ZY, U)−1

2g_p(∇YZ− ∇ZY,∇XU) and similar equalities forg_p((∇YA) (Z, X), U) and g_p((∇ZA) (X, Y), U).

As a consequence,

g_p((∇XA) (Y, Z), U) +g_p((∇YA) (Z, X), U) +g_p((∇ZA) (X, Y), U) =

= 1

2g_p(∇X∇YZ− ∇Y∇XZ, U) +1

2g_p(∇Z∇XY − ∇X∇ZY, U) +1

2g_p(∇Y∇ZX− ∇Z∇YX, U).

Now, [X, Y]_p = [X, Y]^⊥_p ,[X, Z]_p = [X, Z]^⊥_p and [Y, Z]_p = [Y, Z]^⊥_p imply gp�

∇[X,Y]Z, U�

= gp

� T_[X,Y]⊥

pZp, Up

�

=−gp

� T_[X,Y]⊥

pUp, Zp

�

= −g_p�

T_Up[X, Y]^⊥_p , Z_p�

=g_p�

T_UpZ_p,[X, Y]^⊥_p�

= 2gp(TUZ, AXY), as well as

g_p�

∇[Z,X]Y, U�

= 2g_p(T_UY, A_ZX) and

g_p�

∇[Y,Z]X, U�

= 2g_p(T_UX, A_YZ). Applying Bianchi’s identity yields

g_p((∇XA) (Y, Z), U) +g_p((∇YA) (Z, X), U) +g_p((∇ZA) (X, Y), U) =

=g_p(T_UZ, A_XY) +g_p(T_UY, A_ZX) +g_p(T_UX, A_YZ). Finally, we conclude

g_p(R(X, Y)Z, U) = g_p�

∇X∇YZ− ∇Y∇XZ− ∇[X,Y]Z, U�

= Xpg(A_YZ, U)−gp(A_YZ,∇XU)−Ypg(A_XZ, U) +gp(AXZ,∇YU)−gp�

∇[X,Y]Z, U�

= g_p(∇X(A_YZ)− ∇Y (A_XZ), U)−2g_p(T_UZ, A_XY)

= −g_p((∇ZA) (X, Y), U)−g_p(T_UZ, A_XY) +g_p(T_UY, A_ZX) +g_p(T_UX, A_YZ).

Formula 6. Let X, Y, Z and Z^� be horizontal. Then g�

R(X, Y)Z, Z^��

= g�

R^�(X, Y)Z, Z^�� + 2g�

A_XY, A_ZZ^�� +g�

A_XZ, A_YZ^��

−g�

A_YZ, A_XZ^�� ,

where R^�(X_p, Y_p)Z_p denotes the horizontal lift of Rˆ(dπ_pX_p, dπ_pY_p)dπ_pZ_p at every p∈M.