Riemannian Submersions

In this sense,τ^R₁ is associated with the´Hesspµqby the two aforementioned propo-sitions.

3.4.4 Forms onFr_SOpµ^´1p0qq

Recall that the torsion free connection ˜ϕ^Rdecomposes into two one formsφ^R₁ andφ^R₂. φ^R₁ with values in sop4n`kqinduces a connection onFr_SOpµ^´1p0qq, because

φ^R₁ppDk^R₁q^´1p0qq “0, (3.43) R^˚_gφ^R₁ “φ^R₁ @gPOp3kq ĂOp4mq, (3.44) which is true becauseDk^R₁: sop4n`kq ‘sop3kq Ñsop4n`kqis the projection. It allows us to define

ϕ^µ´1p0qpηq “φ^R₁pη˜q, η˜ P pDk^R₁q^´1_q pηq, (3.45) i.e. k^R₁^˚ϕ^µ´1p0q “φ^R₁. Since the solder form onµ^´1p0q pulled back to Fpµ^´1p0q,Mq is the form ˜θ^R, we get the equation

dθ^µ´1p0q,R`^ϕµ´1p0q^^θµ´1p0q,R“⁰, (3.46) and see thatϕ^µ´1p0qis the unique Levi Civita connection onµ^´1p⁰q.

3.4.5 Riemannian Submersions

The next step involves understanding Riemannian submersions on the level of frame bundles. Since there is no exposition of this known to the author, we will describe it in a general setting, and apply it to the reduction afterwards.

Let us at this point recall the basics of the Riemannian submersion theory of O’Neill [O’N66]. A Riemannian submersion π: M^m Ñ ^Bb is a smooth map between two Riemannian manifolds such thatπis a submersion andDπx|H_x: Hx ÑT_πpxqBis a isometry for allxPM, whereHxis the orthogonal complement of kerpDπq ĂTxM.

To such a Riemannian submersion we may associate two important p²,¹q-tensor fields onM,

TXY“H^∇M_VXV^Y`V<sup>∇M</sup><sub>V</sub>XH<sup>Y</sup> (3.47) AXY“H<sup>∇M</sup><sub>H</sub>XV<sup>Y</sup>`V^∇M_HXH^Y, (3.48) where H and V are the horizontal and vertical projection in T M, respectively. T is known to be the second fundamental form of each fiber (if vertical vector fields are plugged in), whereasA is related to the obstruction to integrability of the horizontal distribution onM. An important fact is that

A_XY“ ¹

2V^[X,Y], (3.49) for horizontal vector fieldsX andY. If the Riemannian submersion π: MÑ Bshould also happen to be a principal bundle, and we fix the connection corresponding to the horizontal subspaces, then 2AXY “ ´RpX,Yq, where RpX,Yq is the curvature of the connection, if we identify the vertical tangent space with the Lie algebra as usual.

In the world of principal bundles this can be expressed the following way. Let FrpMqbe the principal bundle of frames andFrpB,Mqthe reduction to adapted frames on M. Here a frame is adapted if it respects the splitting of T Minto horizontal and vertical parts, i.e.

FrpB,Mq “{pPFrpMq:impp|_Rbqis horizontal}. (3.50) Then a pull back of the Levi Civita connection φon FrpMq and the solder formθ gives, after a suitable projection, a connectionψonFrpB,Mqwith structure equation

dθ¹`ψ^θ<sup>1</sup>`τ^θ¹“0, (3.51) where θ¹ is the pull back of the solder form, ψ the projected connection and τ “ i^˚φ´ψ, where i: FrpB,Mq ÑFrpMq is the inclusion. We see that τ is an obstruction to the integrability of the horizontal distribution, because for a product manifold^M“ M1ˆM2 we have the commutative diagram

FrpMq

FrpM1q FrpM1,M2q FrpM2q

M1 M M2

and the connection onFrpMqreduces to a connection onFrpM₁,M₂q, which is the sum of the connections pulled back fromFrpMiq. On the other hand, from the construction of the last section, we also know thatτis related to the second fundamental forms of the fibers.

The notion of horizontal and vertical projection extends to horizontal forms on FrpB,Mq, via

τ_hpξq “τpH^Dπ1pξqq (3.52) τvpξq “τpV^Dπ1pξqq, (3.53) whereπ¹ is the principal bundle map ofFrp^B,Mqand the over line is a lift with respect to that map. It is easy to see that this is well defined for a horizontal form, since it does not depend on the choice of lift. Note also that by definition τ “ τh`τv. The following proposition is the main result of this section.

Proposition3.17(O’Neill on Principal Bundles). τ_vcorresponds toT andτ_hcorresponds toA.

Proof. Note that τ is described by the difference of the connection on FrpMq and the connection onFrpB,Mq. The connection onFrpMqgives rise to the covariant derivative

∇^M, and the connection onFrpB,Mq to ˜∇. As we have shown before, the connection extended from ˜∇splits into two connections which are the Levi Civita connection on the fibers and the horizontal submanifolds, if they exist. Even if they do not, a quick inspection of equation (3.38), using the matrix form of the reduced connection, shows that

∇˜ξX“H^∇Mξ X (3.54) ifξandXare horizontal and

∇˜ηY“V^∇Mη Y, (3.55) ifηandYare vertical. The unique extension of this toFrpMqgives the connection

∇ˆ_χZ:“H^∇Mχ H^Z`V^∇Mχ V^Z, (3.56) for χ an arbitrary tangent vector and Z an arbitrary vector field onM. This can be verified by showing that the above is indeed a covariant derivative on M and that it restricts to ˜∇ if both χ and Z are vertical, or both are horizontal. The latter is immediately clear, the former some simple calculations.

We see now, that

∇^M_χ Z“H^∇M_HχH^Z`H<sup>∇M</sup><sub>V</sub>χH<sup>Z</sup>`H^∇M_HχV^Z`H^∇M_VχV^Z (3.57)

`V<sup>∇M</sup><sub>H</sub>χH<sup>Z</sup>`V^∇M_VχH^Z`V<sup>∇M</sup><sub>H</sub>χV<sup>Z</sup>`V^∇M_VχV^Z

“^AχZ`<sup>T</sup>χZ`_∇ˆ_χZ,

hence the difference of connections indeed gives A`T. Finally, notice that if χ is horizontal then T vanishes, as does τv. If on the other hand χ is vertical, then A

vanishes, as doesτh.

The principal bundle of frames FrpBqof Bcan be pulled back toMviaπ. The Levi Civita connectionφ^B on Frp^Bq can also be pulled back to a connection ˜φ on π^˚Frp^Bq together with the structure equation

φ˜ `_θ˜_B^_φ˜ “0, (3.58) where ˜θBis the pull back of the solder formθBonFrpBq. If we pull this solder form into FrpB,Mq, we get a formθ¹_B, where the obvious restriction map is usedk: FrpB,Mq Ñ π^˚Frp^Bq. A calculation similar to that in remark (3.9) shows that θ¹_B agrees with the part ofθ¹, that has values inR^b. If we split θ¹ into two parts, θ1 and θ2 with values in R^b andR^m´b, andψintoψ1 andψ2 with values in sopbqandsopm´bq, then the structural equation (3.51) ofψdecomposes into

dθ1`ψ1^θ1`τ^θ2“0 (3.59) dθ2`ψ2^θ2`τ^θ1“0. (3.60) If we restrict the first equation toπ-horizontal vectors, the last term vanishes and we see thatψ1 is the Levi Civita connection pulled back from B. Such a restriction also turnsτintoτhand we get the formula

k^˚π^˚φ^B`τh“i^˚φM, (3.61) on FrpB,Mq, if we restrict to vectors lifted from B. This is the recovery of O’Neill’s formula for the connections [O’N66, Lemma3.4].

3.4.6 Forms onFrSOpN,µ^´1p0qq

Applying the last section to the reductionFr_SOp^N,µ^´1p⁰qqof Fr_SOp^µ´1p⁰qqon µ^´1p⁰q, we get the equation

j^R₃^˚ϕ^µ´1p0q“ψ₁`ψ<sub>2</sub>`τ¹, (3.62) whereψ1is the pull back of the Levi Civita connection onN.

3.4.7 Forms onFr_SppN,Mq

Now we will do a similar construction on the quaternionic side of the reduction for Fr_SppN,Mq As withFr_SOpµ^´1p0q,Mq, Fr_SppN,Mq will in general not be horizontal in ι^˚FrSppMq. Using Proposition3.7, we construct a connection ˜ϕ^H with the decomposi-tion

spp^mq “^sppⁿq ‘^op^kq ‘^f, (3.63)

induced by an inclusion ofSppnq ˆSOpkqinSppmqas described in the beginning. As before, the obvious choice of complement will satisfy the necessary condition (3.7).

We get the projected connection form ˜ϕ^Hwhich decomposes into two equivariant one-forms φ^H₁ and φ^H₂ with values in sppnq and sopkq respectively and a difference formτ^Hwith

φ^H₁ `φ<sup>H</sup><sub>2</sub> `τ^H“j^H₂^˚ϕˆ^H. (3.64)