A fundamental differential system of Riemannian geometry

Abstract

We study a fundamental exterior differential system associated to any given
oriented Riemannian manifold M of any dimension. The system was first
considered in hypersurface theory of flat Euclidean space, but here it is
defined invariantly on the tangent sphere bundle of the given Riemannian
manifold. We deduce the structure equations and their main properties. In
particular we write a new equivalent equation for the condition of M being an
Einstein manifold.

Full text

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arXiv:1112.3213v2 [math.DG] 23 Jan 2012
A fundamental differential system of Riemannian geometry
R. Albuquerque∗
rpa@uevora.pt
January 24, 2012
Abstract
We study a fundamental exterior differential system associated to any given oriented
Riemannian manifold M of any dimension. The system was first considered in hypersurface
theory of flat Euclidean space, but here it is defined invariantly on the tangent sphere
bundle of the given Riemannian manifold. We deduce the structure equations and their
main properties. In particular we write a new equivalent equation for the condition of M
being an Einstein manifold.
Key Words: tangent sphere bundle, exterior differential system, Euler-Lagrange systems
MSC 2010: Primary: 58A15, 58A32; 53C21; Secondary: 53C17, 53C25, 53C38, 53C42
The author acknowledges the support of Funda¸ c˜ ao Ciˆ encia e Tecnologia, Portugal, through
CIMA-U´E, Centro de Investiga¸ c˜ ao em Matem´ atica e Aplica¸ c˜ oes da Universidade de´Evora.
1 Introduction
It is a remarkable feature of differential geometry that so many problems have been addressed
through the exterior algebra of differential forms and partial differential equations, the theory
of exterior differential systems. In the present article we are particularly interested in contact
geometry, a branch which interacts strongly with Riemannian geometry. We start by recalling
the well-known contact differential system, generated by a natural non-vanishing 1-form θ, on
the radius s > 0 tangent sphere bundle SMs−→ M of a given smooth oriented Riemannian
n + 1-dimensional manifold M. Then we turn to our main purpose, which is to present the
∗Departamento de Matem´ atica da Universidade de´Evora and Centro de Investiga¸ c˜ ao em Matem´ atica e
Aplica¸ c˜ oes (CIMA-U´E), Rua Rom˜ ao Ramalho, 59, 671-7000´Evora, Portugal.
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study of a set of natural n-forms α0,α1,...,αnexisting always on SMs, apparently known to
a few geometers but which the author rediscovered independently.
Each of these n-forms on the contact 2n + 1-manifold (SMs,θ) and their C∞linear com-
binations assume the natural role of Lagrangian forms. So they induce variational principles
of the underlying exterior differential system. We believe the study of the Lagrangians αi,
0 ≤ i ≤ n, may be pursued through many fields. To be more explicit in this first survey,
suppose we have a (Cartan) coframe of SM1, e0,e1,...,en,en+1,...,e2n, where θ = e0, the
1-forms e0,e1,...,enare horizontal and the remaining are vertical (this is the usual terminol-
ogy of fibre bundle tangent structure and will be explained in section 1). Then in case n = 1,
we have a global coframing (which was already used by Darboux) of θ plus the two 1-forms
α0= e1and α1= e2. In case n = 2,
α0= e12,α1= e14+ e32,α2= e34. (1.1)
In case n = 3,
α0= e123,α1= e126+ e234+ e315,α2= e156+ e264+ e345,α3= e456. (1.2)
In this article we show how these n-forms relate to certain calibrated geometries and, at
least, to one special Riemannian geometry.The latter consists of a natural G2 structure
existing always on SM1for any oriented 4-manifold M. Discovered in [AS09, AS10], it brings
new insight on the role of Einstein metrics.
Here we prove a new Theorem, in a general framework, whose content is as follows: an
oriented Riemannian n + 1-manifold is Einstein if and only if αn−2is coclosed. We trust this
may be quite useful in the field of Einstein metrics.
For instance, for any given metric of constant sectional curvature k we have the formula
dαi= θ ∧ ((n + 1)αi+1− k(n − i + 1)αi−1).(1.3)
This is deduced immediately from Theorem 2.1. Throughout the text the reader will notice
that we try to explore some of the consequences of (1.3). We look forward to develop the
interplay with submanifold theory of space-forms, in a new article, since this seems to be the
a most promising feature of the n-forms.
Further applications of the natural Lagrangians go through an analysis of the Euler-
Lagrange equations of the first few, say when i = 0,1,2, functionals Fi(N) =
set of submanifolds N ֒→ M with liftsˆ N to SM1(as applied by known references in metric
problems of flat ambient space M or in cases of low dimension). We recall in particular the
linear Weingarten equations, which we dare to explore in guidance with the study in [BGG03].
?
ˆ Nαion the
To the best of our knowledge, there exist only a few references about the differential system
of θ and the αi. The treatment is rather distant from that which we propose here (for instance
we define the forms globally from the beginning) and, seemingly, the n-forms have only been
considered having in view the solution of practical problems. Already in Phillip Griffiths’
remarkable work [Gri83] we are presented with the forms in the cases of 2 and 3 dimensional
base M. In [BCG+91] the emphasis is on an application to a metric and algebraic geometry
problem in dimension three (cf.p. 152), the same being true with later articles seen in

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[Gri03]. Coming to more recent consulted works, in [IL03] the n-forms are again perceived
in low dimensional problems. So it seems the general case only appears for the first time in
[BGG03, p. 32], in the discussion of contact differential systems dealing with hypersurface
Riemannian geometry. This extraordinary reference introduces the reader to essentially the
same differential forms αi — it is not difficult to understand they are the same objects we
define here, regardless the Euclidean setting and the underlying or supporting fibre bundle.
The curious formula of the dαifor the case k = 0 in (1.3) is thus already known.
From what we have realized through the literature, for the importance we give to the
subject and the coincidence of the mathematician appearing in all references which mention
explicitly the differential system, the author suggests and uses the name Griffiths forms to
refer to the n-forms αiof the natural exterior differential system of a tangent sphere bundle of
an n + 1-manifold. The latter is henceforth called the Griffiths exterior differential system.
The contents of this article are as follows. In section 2 we present our techniques with the
Riemannian geometry of SMs. We have in view the description of the Griffiths forms and their
first structural equations. In section 3 we present a few applications and examples, namely
to Einstein metrics. There we also look into special Riemannian structures and concentrate
on some variational problems of the geometry of hypersurfaces, a case in which we mostly
follow [BGG03]. In section 4 we call attention upon the study of the infinitesimal symmetries
and conservation laws. Finally in section 5 we complete the more computational proofs from
previous sections.
2 The natural exterior differential system on SMs
2.1Geometry of the tangent sphere bundle
Let M be an n + 1-dimensional Riemannian manifold with metric tensor g = ? , ?. We need
to recall some differential geometry techniques for the study of the total space of the tangent
bundle π : TM −→ M. The metric techniques, studied below in a second phase, have been
thoroughly used and developed in various works by enumerable mathematicians in the last
five decades, of which the most well-known are presumably S. Sasaki, P. Dombrovsky and O.
Kowalsky. One may include some previous articles of the author in regard to this long and
extensive study. The tangent bundle of a given manifold is indeed a proficuous object in the
geometrical context and thence a tool which conveys all analytical fields of application.
The manifold TM is well-known to be a 2n+2-dimensional oriented manifold. A canonical
atlas arising from any given atlas of M induces a natural isomorphism V = kerdπ ≃ π∗TM.
This agrees clearly fibre-wise with the tangent bundle to the fibres of TM. Only supposing a
linear connection ∇ is given on M, we may say the tangent bundle of TM splits as TTM =
H ⊕ V , where H is a sub-vector bundle (depending on ∇). Clearly the horizontal sub-bundle
H is also isomorphic to π∗TM through the map dπ. We may thus define an endomorphism
B : TTM −→ TTM (2.1)
transforming H in V , completing dπ, and vanishing on the vertical sub-bundle V . There also
exists a canonical vector field ξ over TM defined by ξu= u, ∀u ∈ TM. Note ξ is independent
of the connection and the vector ξulies in the vertical side Vu. Henceforth there exists a unique

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horizontal canonical vector field Btξ ∈ H such that B(Btξ) = ξ. Such vector field is called
the geodesic spray of the connection, cf. [Sak96]. In the sequel we let ∇∗= π∗∇ denote the
pull-back connection on π∗TM and let (·)h, (·)vdenote the projections of tangent vectors onto
their H and V components. Then, ∀w ∈ TTM,
∇∗
The manifold TM also inherits a linear connection, denoted ∇∗, which is just
∇∗⊕ ∇∗
preserving the canonical splitting. Of course, the connecting endomorphism B is parallel for
such ∇∗. The theory tells us that, furthermore, for a torsion-free connection ∇, the torsion of
∇∗is given, ∀v,w ∈ TTM, by
T∇∗(v,w) = R∇∗(v,w)ξ = π∗R∇(v,w)ξ := Rξ(v,w)
where R denotes a curvature tensor (the proof is recalled in section 5). This means the torsion
is vertically valued and only depends on the horizontal vectors. The second identity follows by
tensoriality and in the third we have defined the tensor Rξ∈ Ω2
Now we start using the given metric g of M. We recall the Sasaki metric on TM, also
denoted by g, which is given naturally by the pull-back of the metric on M both to H and V .
The morphism B|: H → V is then an isometry. At this point it is admissible the notation
Btfor the adjoint endomorphism of B. Moreover, we remark J = B − Btis well-known to be
an almost complex structure on TM, proving our manifold is indeed always oriented. Notice
∇g = 0 implies ∇∗g = 0.
From now on we assume the given linear connection ∇ is the Levi-Civita connection of M
(though the theory may be extended to any metric connection with torsion).
wξ = wv
andH = ker(∇∗
·ξ). (2.2)
(2.3)
TM(V ).
Finally we are ready to consider the tangent sphere bundle SMswith fixed constant radius
s > 0,
SMs=?u ∈ TM : ?u? = s?.
This hypersurface is also given by the equation ?ξ,ξ? = s2, thence TSMs= ξ⊥⊂ TTM. Since
the manifold TM is orientable, SMsis also always orientable (the restriction of ξ/?ξ? being a
unit normal). Moreover, for any u ∈ TM\0, we may find a local horizontal orthonormal frame
e0,...,en, on a neighbourhood of u ?= 0, such that e0= Btξ/?ξ? (this relies on the smoothness
of the Gram-Schmidt process). Note that any frame in H extended with its mirror in V clearly
determines an orientation on the manifold TM. We adopt the order ‘H then V ’, which makes
a difference with its reverse when dimM is odd.
(2.4)
We are always going to assume M is oriented (though for most purposes the mere existence
of a parallel n + 1-form is sufficient). We let α denote the n-form on TM which is defined
as the interior product of ξ/?ξ? with the vertical pull-back of the volume form of M. We
let vol denote the (usual) pull-back by π of the volume form of M. With the dual coframing
{e0,e1,...,en}, where e0= e♭
mirror subset {ξ♭
TM:
Vol = e0∧ e1∧ ··· ∧ en∧
0, clearly the identity vol = e0∧ ··· ∧ enis verified. Adding the
?ξ?,en+1,...,e2n}, with en+i= ei◦ Bt, ∀i ≥ 1, we may fix the volume form of
ξ♭
?ξ?∧ en+1∧ ··· ∧ e(2n)=(−1)n+1
s
ξ♭∧ vol ∧ α.(2.5)

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Hence, having chosen + or −ξ/s as unit normal direction (according with n odd or even), the
canonical orientation (±ξ/s)?Vol of the Riemannian submanifold SMsagrees with vol ∧ α =
e01···(2n)= e0∧ e1∧ ··· ∧ e(2n). We shall assume always the canonical orientation vol ∧ α on
SMs(the same symbols denote the restriction of those forms to the submanifold in cause). A
direct orthonormal frame as the one introduced here will be said to be adapted.
To facilitate notation we let SM = SMs with any freely chosen s, only recalled when
necessary.
The manifold SM admits a metric linear connection ∇⋆, as we shall see next. For any vector
fields y,z on SM, the covariant derivative ∇∗
to ξ, we just have to add a correction term:
yz is well-defined and, admitting y,z perpendicular
∇⋆
yz = ∇∗
yz −1
s2?∇∗
yz,ξ?ξ = ∇∗
yz +1
s2?yv,zv?ξ.(2.6)
Since ?Rξ(y,z),ξ? = 0, we see from (2.3) that a torsion-free connection D is given by Dyz =
∇⋆
one may further consult [Alb10, Alb11] for details on metric connections on SM.
yz −1
2Rξ(y,z). The reader should be aware D is not the Levi-Civita connection if R∇?= 0;
2.2The contact structure and new n-forms
Continuing to explore the ideas and notation introduced above, we let θ denote the 1-form on
SM
θ = se0= (Btξ)♭= ?ξ,B(·)?.
We wish to make further claims on the natural geometry of SM. They are based on the
following Proposition, whose proof shall be recalled in section 5. The result was essentially
deduced by Y. Tashiro in the late 1960’s through chart computations, cf. [Bla02].
(2.7)
Proposition 2.1. We have dθ = e(1+n)1+ ··· + e(2n)n. Equivalently, ∀v,w ∈ TSM,
dθ(v,w) = ?v,Bw? − ?w,Bv?.(2.8)
It is easy to see that (SM,θ) is a contact manifold. The same is to prove that θ ∧(dθ)n=
(−1)
abbreviation of the wedge product, which happens only when there seems no danger of being
misled. We also observe here that the expression of dθ is not linear in s, lest the reader should
be driven to conclude otherwise.
n(n+1)
2
n!svol ∧ α ?= 0, as we shall care to establish later. We ask the reader to accept our
Remark. We may also describe a metric contact structure on SM.
weights on the fixed metric, the 1-form θ and the so-called Reeb vector field, which is of course
a multiple of Btξ, gives
Finding the correct
ˆ g =
1
4s2g,
ˆξ = 2Btξ,η = ˆ g(ˆξ, ·) =
1
2s2θ,ϕ = B − Bt− 2ξ ⊗ η. (2.9)
And then we have η(ˆξ) = 1, ϕ(ˆξ) = 0, ϕ2= −1 + η ⊗ˆξ, ˆ g(ϕ·,ϕ·) = ˆ g − η ⊗ η and
dη = 2ˆ g(·,ϕ·) as we wished. This metric contact structure is Sasakian if and only if M has
constant sectional curvature 1/s2.