A Characterisation of Smooth Maps into a Homogeneous Space
We generalize Cartan's logarithmic derivative of a smooth map from a manifold into a Lie group to smooth maps into a homogeneous space = /, and determine the global monodromy obstruction to reconstructing such maps from infinitesimal data. The logarithmic derivative of the embedding of a subm...
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| description | We generalize Cartan's logarithmic derivative of a smooth map from a manifold into a Lie group to smooth maps into a homogeneous space = /, and determine the global monodromy obstruction to reconstructing such maps from infinitesimal data. The logarithmic derivative of the embedding of a submanifold Σ ⊂ becomes an invariant of Σ under symmetries of the ''Klein geometry'' , whose analysis is taken up in [SIGMA 14 (2018), 062, 36 pages, arXiv:1703.03851].
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 029, 15 pages
A Characterisation of Smooth Maps
into a Homogeneous Space
Anthony D. BLAOM
University of Auckland, New Zealand
E-mail: anthony.blaom@gmail.com
URL: https://ablaom.github.io
Received June 25, 2021, in final form April 04, 2022; Published online April 10, 2022
https://doi.org/10.3842/SIGMA.2022.029
Abstract. We generalize Cartan’s logarithmic derivative of a smooth map from a manifold
into a Lie group G to smooth maps into a homogeneous space M = G/H, and determine the
global monodromy obstruction to reconstructing such maps from infinitesimal data. The
logarithmic derivative of the embedding of a submanifold Σ ⊂ M becomes an invariant of Σ
under symmetries of the “Klein geometry” M whose analysis is taken up in [SIGMA 14
(2018), 062, 36 pages, arXiv:1703.03851].
Key words: homogeneous space; subgeometry; Lie algebroids; Cartan geometry; Klein geo-
metry; logarithmic derivative; Darboux derivative; differential invariants
2020 Mathematics Subject Classification: 53C99; 22A99; 53D17
1 Introduction
According to a theorem of Élie Cartan, a smooth map f : Σ → G, from a connected manifold Σ
into a Lie group G, is uniquely determined by its logarithmic derivative, up to right translations
in G. This derivative, also known as the Darboux derivative of f , is a one-form δf on Σ
taking values in the Lie algebra of g of G. Here we formulate and prove a generalization of this
result, Theorem 3.14, to smooth maps f : Σ → G/H into an arbitrary homogeneous space G/H.
Our generalization describes explicitly the global obstruction to reconstructing such maps from
infinitesimal data, data that generalizes logarithmic derivatives (generalized Maurer–Cartan
forms).
In this introduction we generalize the notion of Maurer–Cartan forms and their monodromy,
and state the main existence Theorem 1.5. The proof is straightforward, apart from a question
about Lie algebroid integrability, which is addressed in Section 2, by applying [5]. Uniqueness,
up to symmetry, is guaranteed under a mild topological condition on G/H, but we must take
some care to qualify what is meant by “symmetry”, a task postponed to Section 3.
Cartan’s theorem is commonly associated with his method of moving frames for studying sub-
geometry. While the moving frames method can be reinterpreted within the present framework,
it is possible to study subgeometry using the new theory without fixing frames or local coordi-
nates. Both frame and frame-free illustrations are given in a sequel article [1]. It is instructive
to review Cartan’s approach here. For more detail we recommend [8].
Cartan’s method of moving frames
To classify, with a unified approach, the submanifolds of Euclidean space, affine space, confor-
mal spheres, projective space, and so on, the ambient space M is viewed as a homogeneous
space G/H, i.e., as a “Klein geometry”. Here G is the group of symmetries of the geometric
structure on M , which acts transitively by assumption.
mailto:anthony.blaom@gmail.com
https://ablaom.github.io
https://doi.org/10.3842/SIGMA.2022.029
2 A.D. Blaom
Using the group structure, one tries to replace the embedding f : Σ → G/H of a subma-
nifold Σ with certain data defined just on Σ and amounting to an infinitesimalization of the
map f . The infinitesimal data consists of invariants of f — that is, the data depends on f only
up to symmetries of G/H (G-translations). However, these invariants ought to be complete,
in the sense that they are sufficient for the reconstruction of f , up to symmetry.
Cartan’s method for finding a complete set of invariants is in two steps. In the first step one
attempts to lift the embedding f : Σ → M ∼= G/H to a smooth map f̃ : Σ → G:
G
Σ G/H.
f̃
f
The lift, which is not unique, should be as canonical as possible, to make the identification of
invariants easier later on. For example, given a curve in Euclidean three-space f : [0, 1] → R3,
one obtains a lift f̃ : [0, 1] → G into the group of rigid motions by declaring f̃(t) to be the
rigid motion mapping the Frenet frame of the curve at f(0) to the Frenet frame at f(t) — the
“moving frame”.
Now the basic infinitesimal invariant of a Lie group G is the Maurer–Cartan form, a one-
form on G taking values in its Lie algebra g. In the second step of Cartan’s procedure, one pulls
the Maurer–Cartan form back from G to a one-form δf̃ on Σ using the lifted map f̃ : Σ → G.
By Cartan’s theorem recalled below, one can reconstruct f̃ : Σ → G, and hence the map f : Σ →
G/H, from a knowledge of δf̃ alone, which accordingly encodes (indirectly) complete invariants
for the embedding.
Smooth maps into a Lie group
Fix a Lie group G and let g denote its Lie algebra. A Maurer–Cartan form on a smooth
manifold Σ is a g-valued one-form ω satisfying the Maurer–Cartan equations,
dωk +
∑
i<j
cijk ωi ∧ ωj = 0,
where the ωk are components of ω with respect to some basis of g, and cijk the corresponding
structure constants. We have written the Maurer–Cartan equations as they are most commonly
recognized, although this is not best representation from the present point of view, as we shall see.
The Lie group G itself supports a unique right-invariant Maurer–Cartan form ωG that is the
identity on TeG = g. Every smooth map f : Σ → G pulls ωG back to a Maurer–Cartan form
on Σ, here denoted δf . Since
δf
(
d
dt
x(t)
)
=
d
dτ
f(x(τ))f(x(t))−1
∣∣∣∣
τ=t
,
or δf = d log(f) in the special case G = (0,∞), δf is called the logarithmic derivative of f .
Theorem 1.1 (Cartan). Every Maurer–Cartan form ω on a simply-connected manifold Σ is
the logarithmic derivative of some smooth map f : Σ → G. If f ′ : Σ → G is a second map with
logarithmic derivative ω, then there exists a unique g ∈ G such that f ′(x) = f(x)g.
One says that f is a primitive of ω. If Σ is only connected, then the obstruction to the
existence of a primitive is known as the monodromy. Anticipating our later generalization, we
recall two forms of the monodromy here. For further details see, e.g., [10, Theorem 7.14, p. 124].
A Characterisation of Smooth Maps into a Homogeneous Space 3
The global form of the monodromy is a groupoid morphism
Ω: Π(Σ) → G, (1.1)
where Π(Σ) is the fundamental groupoid Π(Σ) of Σ. By definition, an element of [γ] ∈ Π(Σ)
is the homotopy equivalence class of a path γ : [0, 1] → Σ (endpoints fixed). Since the interval
[0, 1] is simply-connected, the Maurer–Cartan form γ∗ω on [0, 1] admits, by Cartan’s theorem,
a unique primitive Γ: [0, 1] → G satisfying Γ(0) = 1G, known as the development of ω along the
path γ. One shows that Γ(1) depends only on the class [γ] and one defines Ω([γ]) = Γ(1).
If ω is the logarithmic derivative of some map f : Σ → G, then Ω([γ]) = f(γ(1))f(γ(0))−1.
In particular, fixing some x0 ∈ Σ,
f(x) = Ω([γ])f(x0), (1.2)
where γ : [0, 1] → Σ is any path from x0 to x. If ω is an arbitrary Maurer–Cartan form, then we
attempt to define a primitive f : Σ → G by (1.2). The group of all elements of Π(Σ) beginning
and ending at x0 is the fundamental group π1(Σ, x0) and f is well-defined if the restriction
of Ω to a group homomorphism Ωx0 : π1(Σ, x0) → G — which we call the pointed form of
the monodromy — is trivial, i.e., takes on the constant value 1G. This condition is evidently
independent of the choice of fixed point x0.
Complete invariants without lifts
Global lifts as described above do not exist in general and Cartan’s method has been largely
limited to the local reconstruction of smooth maps into a homogeneous space, and the special
case of curves (dimΣ = 1). This is despite the fact that Theorem 1.1 and the monodromy
obstruction are global results!
A generalization of Theorem 1.1 to smooth maps f : Σ → G/H obviates the need for lifts.
Specifically, what we present here is a characterization of smooth maps f : Σ → M , where M
is an arbitrary space on which some Lie group G is acting transitively, a subtle but significant
change in viewpoint, as we shall explain in Section 3. Our results are naturally formulated in
the language of Lie algebroids, and the proof is an application of Cartan’s fundamental theorems
for Lie groups, known as Lie I, Lie II and Lie III, generalized to Lie groupoids, with which we
will assume some familiarity (see, e.g., [5, 6]). Standard introductions to Lie groupoids and
algebroids are [4, 6, 7, 9].
Logarithmic derivatives
In Lie algebroid language, a Maurer–Cartan form on Σ is nothing more than a morphism
ω : TΣ → g of Lie algebroids, and Theorem 1.1 a special case of Lie II, as is well-known. In the
general setting, we replace the Maurer–Cartan form on G with the action algebroid g×M asso-
ciated with the action of G on M , and use f : Σ → M to pull g×M back to a Lie algebroid A(f)
over Σ. Of course this pullback must be performed in the category of Lie algebroids rather than
vector bundles (see, e.g., [9, Section 4.2]). The composite δf : A(f) → g of the natural maps
A(f) → g×M → g is a Lie algebroid morphism, which becomes the logarithmic derivative of f .
The results to be described here show that δf — or more precisely an appropriate equivalence
class of δf , see Section 3 — is a complete invariant of f .
Example 1.2 (logarithmic derivative of an embedding). Suppose Σ ⊂ M is a submanifold
and f : Σ → M the embedding. Then A(f) is the subbundle of the trivial bundle g × Σ → Σ
consisting of all pairs (ξ, x) having the property that the integral curve on M through x ∈ Σ
4 A.D. Blaom
of the infinitesimal generator ξ† of ξ ∈ g is tangent to Σ ⊂ M at x. The anchor of A(f) is
(ξ, x) 7→ ξ†(x) and the bracket well-defined by
[X,Y ] = ∇#XY −∇#Y X + {X,Y }.
Here ∇ is the canonical flat connection on g × Σ and, viewing sections of g × Σ as g-valued
functions, {X,Y }(x) := [X(x), Y (x)]g. The logarithmic derivative δf is the composite A(f) ↪→
g× Σ → g.
Generalized Maurer–Cartan forms
Again let M be a smooth manifold on which some Lie group G is acting from the left transi-
tively — what is hereafter referred to as a homogeneous G-space. With this data fixed, our next
task is to introduce axioms for Lie algebroid morphisms ω : A → g, where A is a Lie algebroid
over some manifold Σ, modeled on local properties of logarithmic derivatives δf of smooth maps
f : Σ → M .
To this end, observe that logarithmic derivatives map Lie algebroid isotropy algebras isomor-
phically onto isotropy algebras of the action of G on M . Specifically, if we denote the kernel
of an anchor map #: A → TΣ by A◦, then, for any x ∈ Σ, A(f)◦x and gf(x) have the same
dimension, and
δf(A(f)◦x) = gf(x). (1.3)
The following axioms, then, are no stronger than properties already satisfied by logarithmic
derivatives:
M1 A is transitive.
M2 For some point x0 ∈ Σ, the restriction ω : A◦
x0
→ g is injective.
M3 For some (possibly different) point x0 ∈ Σ, there exists m0 ∈ M such that ω
(
A◦
x0
)
⊂ gm0 ,
which will be written x0
ω−→ m0.
Using the shorthand defined in M3, (1.3) reads x
δf−→ f(x). A Lie algebroid morphism ω : A → g
is called a generalized Maurer–Cartan form if it satisfies M1–M3.
Contrary to the group case, a Lie algebroid is not necessarily integrable (the Lie algebroid of
some Lie groupoid) and obstructions to integrability are subtle. See [6] for a fuller discussion
and examples. Nevertheless, we have:
Proposition 1.3. Assume M1 and M2 hold and that Σ is connected. Then:
1. A is an integrable Lie algebroid.
2. M2 holds with x0 replaced by an arbitrary point x ∈ Σ.
3. If M3 holds, then it holds with x0 replaced by an arbitrary point x ∈ Σ (and suitable choice
of replacement m ∈ M for m0).
We shall see the first assertion readily implies the others. A simple proof of (1) is not known
to us.1 In Section 2, where the proposition is proven, we will easily deduce integrability from
Crainic and Fernandes’ generalization of Lie III [5].
1In an earlier version of this manuscript the main existence theorem was proven without assuming integrability,
using substantially more complicated arguments, and integrability established post facto.
A Characterisation of Smooth Maps into a Homogeneous Space 5
Principal primitives
In fact, the most näıve notion of a primitive is not unique “up to symmetry”. However, the
näıve notion will play a role and be given a name:
Definition 1.4. A smooth map f : Σ → M is a principal primitive of a generalized Maurer–
Cartan form ω : A → g if there exists a Lie algebroid morphism L : A → A(f) such that the
following diagram commutes:
A g
A(f).
L
ω
δf
(1.4)
Note that we do not assume L is an isomorphism, or even that A and A(f) have the same
rank.
Monodromy obstructions to the existence of primitives
We now offer this paper’s main construction, and formulate the existence part of our results.
Let M be a homogeneous G-space and ω : A → g an associated generalized Maurer–Cartan
form. We are going to explicitly describe the obstruction to the existence of a principal prim-
itive f : Σ → M of ω, where Σ is the base of A. The most natural description is in terms of
some abstract transitive Lie groupoid G integrating A, whose existence is guaranteed by Propo-
sition 1.3, and which we may take to be source-simply connected, on account of Lie I. In the
next subsection we will offer a more concrete interpretation using a generalization of Cartan’s
development along paths.
According to Lie II, ω is the derivative of a unique Lie groupoid morphism
Ω: G → G, (1.5)
which we call the global form of the monodromy of ω, being the analogue of (1.1). Continuing
the analogy, we choose x0 ∈ Σ and m0 such that M2 and M3 hold, and attempt to define
a principal primitive f : Σ → M , mapping x0 to m0, by f(x) = Ω(p) ·m0, where p ∈ G is any
arrow from x to x0. In this ambition we are successful, so long as f is well-defined, i.e., provided
Ω(Gx0) ⊂ Gm0 . (1.6)
Here Gx0 ⊂ G denotes the group of all arrows p ∈ G beginning and ending at x0, and Gm0 the
isotropy at m0 of the action of G on M .
Now the algebroid isotropy A◦
x0
is the Lie algebra of Gx0 and, by our hypothesis M3, ω
(
A◦
x0
)
⊂ gm0 . Therefore,
Ω
(
G◦
x0
)
⊂ Gm0 , (1.7)
where G◦
x0
is the connected component of Gx0 . Moreover, as the transitive Lie groupoid G has
simply-connected source-fibres, there is a natural exact sequence
1 → G◦
x0
→ Gx0
ρ−→ π1(Σ, x0) → 1.
From this and (1.7) we obtain a map Ωm0
x0
: π1(Σ, x0) → M well-defined by
Ωm0
x0
(ρ(p)) = Ω(p) ·m0.
We call this the pointed form of the monodromy. By construction, our requirement (1.6) is
equivalent to Ωm0
x0
taking a constant value (which is necessarily m0).
6 A.D. Blaom
Theorem 1.5 (existence and uniqueness of principal primitives). A Maurer–Cartan form ω:
A → g over a connected manifold Σ admits a principal primitive f : Σ → M if and only if the
pointed form of the monodromy Ωm0
x0
: π1(Σ, x0) → M is constant for some (and consequently
any) choice of x0 and m0 with x0
ω−→ m0. In that case there is a unique principal primitive f
of ω such that f(x0) = m0.
Proof. The preceding arguments establish the existence of a primitive, given constant mon-
odromy. Conversely, given the existence of a primitive f with m0 = f(x0), one easily establishes
constancy of the monodromy. For example, an elementary observation stated later as Proposi-
tion 3.12 shows that
f(x) = Ω(x0) ·m0 (1.8)
for any arrow p ∈ G from x0 to x and so, in particular, m0 = Ω(p) · m0 for any p ∈ Gx0 .
Since (1.8) applies to any principle primitive with m0 = f(x0), the last statement of the theorem
also holds. ■
Monodromy as development along A-paths
Let A be a Lie algebroid over a connected manifold Σ. Then a piece-wise smooth map a : [0, 1]
→ A, covering an ordinary path γ : [0, 1] → Σ, is called an A-path if #a(t) = γ̇(t), for all
t ∈ [0, 1]. Here #: A → TΣ denotes the anchor of A.
Every A-path a can be understood as Lie algebroid morphism â : TI → A defined by
â(∂/∂t) = a and all such morphisms arise from A-paths. In particular, given any Lie groupoid G
integrating A, we may apply Lie II, obtaining a Lie groupoid morphism I × I → G. The image
of (0, t) under this morphism, denoted∫ t
0
a ∈ G,
is an arrow from γ(0) to γ(t). If A is a Lie algebra, and G a Lie group with Lie algebra A, then a
is simply a piece-wise smooth path in the Lie algebra, and the integral above the usual integral
to an element in the group. This familiar case is the one applying in the proposition below:
Proposition 1.6. Consider a Maurer–Cartan form ω : A → g as in the preceding theorem,
and suppose x0
ω−→ m0, for some x0 ∈ Σ and m0 ∈ M . Let [γ] ∈ π1(Σ, x0) be given and let
a : [0, 1] → A be any A-path covering γ. Then the monodromy is given by
Ωm0
x0
([γ]) =
(∫ 1
0
ω ◦ a
)
·m0.
Proof. The proposition follows immediately from the definition of Ωm0
x0
and the following ele-
mentary property of A-paths: Every Lie algebroid morphism ω : A1 → A2 maps A1-paths to
A2-paths, and if ω is the derivative of a Lie groupoid morphism Ω: G1 → G2, then, for any
A1-path a,
Ω
(∫ t
0
a
)
=
∫ t
0
ω ◦ a, t ∈ I. ■
Invariants for subgeometry and Bonnet-type theorems
As far as we know, Cartan’s method of moving frames is the only general technique for obtaining
invariants of a submanifold Σ of a Klein geometry M ∼= G/H, and for proving theorems which
A Characterisation of Smooth Maps into a Homogeneous Space 7
reconstruct the submanifold from its invariants (up to symmetry). The fundamental theorem of
surfaces (Bonnet theorem) is a prototype for results of this kind. For the special class of parabolic
geometries (G/H a flag manifold) an approach based on tractor bundles is outlined in [2] and
successfully applied to conformal geometry (see also [3]). These authors do not describe the
monodromy, however, restricting their attention to the case of simply-connected submanifolds.
While the logarithmic derivative δf introduced here delivers a complete invariant of an em-
bedding f : Σ ↪→ G/H, it is usually too abstract to be immediately useful. In [1] we take up the
problem of “deconstructing” this invariant, and offer illustrations to concrete geometries.
Bracket convention
Throughout this article, brackets on Lie algebras and Lie algebroids are defined using right-
invariant vector fields.
2 Integrability
In this section we prove Proposition 1.3 by applying Crainic and Fernandes’ generalization of
Lie III [5].
Let A be a Lie algebroid over Σ. In [5, 6] the kernel of the anchor of A → TΣ is denoted
by g. However, as this conflicts with our use as the Lie algebra of G, we continue to denote the
kernel by A◦. We otherwise follow the notation and terminology of [6].
In particular, the Weinstein groupoid of A is denoted G(A). An element of G(A) is a certain
equivalence class of A-paths. The obstruction to the existence of a bona fide Lie groupoid
integrating A (that is, to the topological groupoid G(A) being a Lie groupoid) is measured by
the monodromy groups Ñx0(A), x0 ∈ Σ. By definition, Ñx0(A) is the kernel of the natural
homomorphism
G
(
A◦
x0
)
→ G(A)◦x0
. (2.1)
On the right-hand side ◦ denotes connected component. At the level of Lie algebroid paths, this
homomorphism is just inclusion. The object on the left is a Lie group, while that on the right
may only be a topological group. Specializing [5] to the transitive case, we have
Theorem 2.1. Assuming its base manifold Σ is connected, the Lie algebroid A is integrable if
and only if Ñx0(A) ⊂ G
(
A◦
x0
)
is discrete, for some x0 ∈ Σ.
Now assume M1 and M2 hold. Then the restriction ω : A◦
x0
→ g is an injection, integrating
to a homomorphism Ω: G
(
A◦
x0
)
→ G of Lie groups, whose kernel K0 is accordingly discrete.
On the other hand, we may include this homomorphism in the following commutative diagram,
whose vertical arrow is (2.1):
G
(
A◦
x0
)
G
G(A)◦x0
.
Ω
Here the diagonal map is the restriction of the natural topological groupoid morphism G(A) →
G(g) = G, i.e., the map sending the equivalence class of an A-path a to the equivalence class of
the g-path ω ◦ a. Commutativity of the diagram implies the kernel of the vertical map must lie
in K0, but this kernel is, by definition, Ñx0(A). This shows Ñx0(A) is a discrete subset of G
(
A◦
x0
)
and the proof of Proposition 1.3(1) now follows from the theorem above.
8 A.D. Blaom
Now let x ∈ Σ be arbitrary and let G be a Lie groupoid integrating A, which is necessarily
transitive. Then, assuming Σ is connected, there exists an arrow p ∈ G from x0 to x. Conjugation
Cp : q 7→ pqp−1 maps the isotropy group Gx0 isomorphically onto Gx. Differentiating, we get a Lie
algebra isomorphism dCp : A
◦
x0
→ A◦
x and a commutative diagram
A◦
x0
dCp−−−−→ A◦
x
ω
y yω
g
AdΩ(p)−−−−→ g.
From this observation parts (2) and (3) of Proposition 1.3 immediately follow.
3 The uniqueness of primitives up to symmetry
This section establishes the uniqueness of primitives, appropriately defined, up to symmetry.
The central result is Theorem 3.9. Combining our uniqueness result with Theorem 1.5, we
obtain the existence and uniqueness Theorem 3.14.
Symmetries of a homogeneous space
For the purposes of constructing a theory with the proper invariance, we have been regarding our
fixed data as a homogeneous G-space. While a choice of point m0 ∈ M gives us an identification
M ∼= G/Gm0 , formulations depending on a choice of base point are to be eschewed.2 This
decision has a somewhat unexpected consequence, anticipated by reconsidering the simplest
case.
According to Cartan’s theorem, a smooth map f : Σ → G is uniquely determined by its
logarithmic derivative, up to symmetries of G. Here a “symmetry” is a right group translation.
However, to obtain an invariant version of Cartan’s result we must broaden both the notion of
symmetry and what it means to be a primitive. To see why, consider a smooth map f : Σ → M ,
where M is a smooth manifold on which G is acting transitively and freely, so that M ∼= G, up
to choice of base-point. In order to drop the right-invariant Maurer–Cartan form ωG on G to
a one-form on M , we must suppose here that G is acting on M from the left. For then, fixing
m0 ∈ M and defining a Φ(g) = g · m0, the diffeomorphism Φ: G → M pushes ωG forward to
a one-form ωM on M that is independent of the choice of m0. But then ωM is not invariant with
respect to the action of G — rather it is equivariant, if we regard G as acting on g by adjoint
action. In particular, two smooth maps f1, f2 : Σ → M with f2(x) = g · f1(x), x ∈ Σ, have,
in general, different logarithmic derivatives: f∗
2ωM = ⟨Adg, f∗
1ωM ⟩.
To proceed one defines f : Σ → M to be a primitive of a one-form ω on Σ if f∗ωM and ω
agree “up to adjoint action”. The price one pays for this relaxed definition is that the loga-
rithmic derivative f∗ωM only determines f up to a larger class of symmetries of M . Under an
identification M ∼= G, these symmetries consist of the diffeomorphisms generated by all right
and left translations.
Symmetries in the general case are formalised as follows:
Definition 3.1. Let M be a homogeneous G-space. Then a symmetry of M is any diffeomor-
phism ϕ : M → M for which there exists some l ∈ G such that ϕ(g · m) = lgl−1 · ϕ(m) for all
g ∈ G, m ∈ M .
2Geometries in the real world do not come with a preferred choice of base-point. Base-points are an artifact
of Klein’s abstraction of geometry, not an intrinsic feature.
A Characterisation of Smooth Maps into a Homogeneous Space 9
The symmetries of M form a Lie group henceforth denoted Aut(M). Evidently, Aut(M)
contains every left translation ϕ(m) := k ·m, k ∈ G (take l = k). The following characterization
of symmetries is readily verified:
Proposition 3.2. Fix a point m0 ∈ M and identify M with the left coset space G/H, where H
denotes the isotropy at m0. Let l ∈ G be arbitrary and suppose r ∈ G is in the normaliser of H,
so that there exists a map ϕ : G/H → G/H making the following diagram commute:
G
g 7→ lgr−1
−−−−−−−→ Gy y
G/H
ϕ−−−−→ G/H.
Then ϕ is a symmetry of M and all symmetries of M arise in this way. In other words, the Lie
group W := NG(H)/H acts on the left of G/H according to
rH · gH = gr−1H,
(an action commuting with the left action of G) and Aut(M) is the Lie group generated by both
left translations and those transformations of M ∼= G/H defined by the action of W . Here
NG(H) denotes the normaliser of H in G.
In contrast to the special case in which G acts freely, Aut(M) is frequently not much larger
than the group G of left translations, in applications of interest to geometers:
Examples 3.3.
1. Take M = Rn, let H ⊂ GL(n,R) be any linear Lie group whose fixed point set is the origin,
and let G ∼= H ⋉Rn be the group of transformations of Rn generated by translations and
elements of H. Then NG(H) = H and accordingly Aut(M) = G.
2. (Affine geometry) As special cases of item (1), we may take G = GL(n,R) or G = SL(n,R)
and obtain the affine and equi-affine geometries, with Aut(M) = G.
3. (Euclidean, elliptic and hyperbolic geometry) Take M to be one of Riemannian space
forms Rn, Sn or Hn, and let G be the full group of isometries. Then in every case it
is possible to show that each element of NG(H)/H has a representative r ∈ NG(H) lying
in the centre of G, and it follows easily that Aut(M) = G.
4. (Special elliptic geometry) Take M = Sn but let G be the group of orientation-preserving
isometries, SO(n+ 1). In this case a little more work reveals that
Aut(M) =
{
SO(n+ 1) if n is odd,
O(n+ 1) if n is even.
That is, for even-dimensional spheres, we must add to G the orientation-reversing isome-
tries to obtain the full symmetry group.
5. Suppose M is a homogeneous G-space where G is compact and connected and has trivial
centre, and suppose that the isotropy subgroup H at some point of M is a maximal torus.
Then NG(H)/H is the Weyl group, well-known to be finite.
6. (Parabolic geometries) For a flag manifold M , such as a conformal sphere or projective
space, G is a connected semi-simple Lie group and the isotropy group H is a parabolic
subgroup of G. In this case also NG(H)/H is known to be finite.
10 A.D. Blaom
Morphisms between Maurer–Cartan forms
Henceforth we drop the qualification “generalized”: All Maurer–Cartan forms and logarithmic
derivatives will be understood in the generalized sense.
With G, M and Σ fixed as in the Introduction (under “Generalized Maurer–Cartan forms”)
we collect all associated Maurer–Cartan forms into the objects of a category. In this category
a morphism ω1 → ω2 between objects ω1 : A1 → g and ω2 : A2 → g consists of a Lie algebroid
morphism λ : A1 → A2 covering the identity on Σ and an element l ∈ G such that the following
diagram commutes:
A1
ω1−−−−→ g
λ
y yAdl
A2
ω2−−−−→ g.
If λ is injective, we will say that ω1 → ω2 is monic. The preceding abstractions are justified by
the following observation (strengthened in special cases in Theorem 3.9 below):
Proposition 3.4. Let f1 : Σ → M be a smooth map into a homogeneous G-space M and define
a second smooth map f2 : Σ → M by f2 = ϕ ◦ f1, for some ϕ ∈ Aut(M). Then δf1 and δf2 are
isomorphic in the category of Maurer–Cartan forms.
That is, smooth maps f1, f2 : Σ → M agreeing up to a symmetry of M have isomorphic
logarithmic derivatives.
Proof. Supposing f2 = ϕ ◦ f1, ϕ ∈ Aut(M), define l ∈ G as in Definition 3.1. Then the map
Adl ×ϕ, defined by
(ξ, x) 7→
(
Adl ξ, ϕ(x)
)
,
g×M → g×M
is a Lie algebroid automorphism of the action algebroid g×M covering ϕ : M → M . In particular,
the composite A(f1) → g ×M
Adl ×ϕ−−−−→ g ×M is a Lie algebroid morphism J sitting in a com-
mutative diagram
A(f1)
J−−−−→ g×My y
TΣ −−−−→
Tf2
TM.
The vertical arrows indicate anchor maps. Explicitly, we have
J(X) =
(
Adl δf1(X), f2(⌟X)
)
,
where ⌟X ∈ Σ denotes the base point of X.
As A(f2) is the pullback of g × M under f2, we obtain, from the universal property of
pullbacks, a unique Lie algebroid morphism λ : A(f1) → A(f2) such that J is the composite
A(f1)
λ−→ A(f2)
δf2−−→ g×M.
This immediately implies commutativity of the diagram
A(f1)
δf1−−−−→ g
λ
y yAdl
A(f2)
δf2−−−−→ g.
A Characterisation of Smooth Maps into a Homogeneous Space 11
One argues that λ is an isomorphism by replacing ϕ with ϕ−1 and reversing the roles of f1
and f2. ■
Primitives
A smooth map f : Σ → M will be called a primitive of the Maurer–Cartan form ω : A → g if
there exists a morphism ω → δf . Evidently, every principal primitive is a primitive.
Maximal Maurer–Cartan forms
Note that Axioms M1 and M2, together with Proposition 1.3, imply the following restrictions
on the necessarily constant rank of A, whenever ω : A → g is a Maurer–Cartan form:
dim g− dimM ⩽ rankA ⩽ dim g− dimM + dimΣ.
We say ω is maximal if A has maximal rank, i.e., if
dim g− rankA = dimM − dimΣ.
In this case it follows from M3, Proposition 1.3, and a dimension count that
M3′. For any point x ∈ Σ there exists m ∈ M such that ω(A◦
x) = gm.
Logarithmic derivatives and ordinary Maurer–Cartan forms are always maximal.
Lemma 3.5. Every morphism ω → δf is monic. In particular, if ω is maximal, then ω → δf
is an isomorphism.
Proof. A morphism ω → δf consists of a Lie algebroid morphism λ : A → A(f) covering the
identity on Σ, and l ∈ G, such that
δf(λ(a)) = Adl ω(a), a ∈ A. (3.1)
Suppose λ(a) = 0, a an element of A with base-point x ∈ Σ. Since λ is a Lie algebroid morphism
covering the identity, we have #a = 0, i.e., a ∈ A◦
x. Since ω(a) = 0, by (3.1), Axiom M2 and
Proposition 1.3 imply a = 0. ■
The existence Theorem 1.5 has the following corollary (of which we make no further use):
Corollary 3.6. Every Maurer–Cartan form ω : A → g with constant monodromy has an exten-
sion to a maximal Maurer–Cartan form ω : A′ → g, for some Lie algebroid A′ ⊃ A.
Proof. By the existence theorem, ω admits a principal primitive f : Σ → M . That is, there ex-
ists a morphism λ :A → A(f), injective by the lemma, whose logarithmic derivative δf :A(f) → g
fits into the commutative diagram (1.4). The logarithmic derivative of f is then a maximal
Maurer–Cartan form extending ω. ■
Uniqueness of primitives
As usual, suppose G acts transitively on M , and let G◦
m0
denote the connected component of
the isotropy Gm0 at some m0 ∈ M . Then since G◦
m0
is path-connected, NG
(
Gm0
)
⊂ NG
(
G◦
m0
)
.
Definition 3.7. We say the isotropy groups of the G action are weakly connected if for some
(and hence any) m0 ∈ M , we have NG
(
Gm0
)
= NG
(
G◦
m0
)
.
12 A.D. Blaom
Example 3.8. If M is one of the Riemannian space forms Rn, Sn or Hn, and G is the full group
of isometries, then although the isotropy groups of the action of G on M are not connected,
they are weakly connected.
A proof of the following central result appears below.
Theorem 3.9. Suppose the action of G on M has weakly connected isotropy groups. Let
f1, f2 : Σ → M be smooth maps. Then there exists an isomorphism δf1 ∼= δf2 in the category of
Maurer–Cartan forms if and only if there exists ϕ ∈ Aut(M) such that f2 = ϕ ◦ f1.
In contrast to the classical setting (Theorem 1.1) there may exist more than one choice of
ϕ ∈ Aut(M) for which f2 = ϕ ◦ f1, even if G acts faithfully on M . For example, consider two
constant maps f1, f2.
Combining the theorem with the lemma above, we obtain:
Corollary 3.10 (uniqueness of primitives). If the action of G on M has weakly connected
isotropy groups then primitives f : Σ → M of a maximal Maurer–Cartan form are unique, up to
symmetries of M .
A non-maximal Maurer–Cartan form may have distinct primitives not related by a symmetry:
Example 3.11. Let G be the group of isometries of the plane M = R2 with Lie algebra g
identified with the Killing fields. Let x, y : R2 → R denote the standard coordinate functions
and let ω : TR → g be the generalized Maurer–Cartan form3 defined by
ω
(
∂
∂t
)
= −y
∂
∂x
+ x
∂
∂y
(a constant element of g).
Then for any r ⩾ 0 the map f(t) = (r cos t, r sin t) is a primitive of ω.
For the proof of the theorem we need one additional observation:
Proposition 3.12. Suppose f : Σ → M is a principal primitive of a Maurer–Cartan form
ω : A → g. Let Ω: G → G be the global form of the monodromy of ω, as defined in (1.5). Then,
for any x ∈ Σ, one has x
ω−→ f(x), and for any x0 ∈ Σ,
f(x) = Ω(p) · f(x0),
where p ∈ G is any arrow from x0 to x.
Proof. For some Lie algebroid morphism λ : A → A(f), we have a commutative diagram
A
ω−−−−→ g
λ
y x
A(f) −−−−→ g×M.
Since λ covers the identity, the claim x
ω−→ f(x) follows easily from commutativity and the
definition of the bottom map. Let G(f) denote the pullback of the action groupoid G×M by f .
Since G is source-simply-connected, λ is the derivative of a Lie groupoid morphism Λ: G → G(f)
3Actually ω is an ordinary Maurer–Cartan form in this case but we are understanding primitives as maps
into R2, not maps into G!
A Characterisation of Smooth Maps into a Homogeneous Space 13
and the following diagram commutes (because the composites being compared have a source-
connected domain and identical derivatives, by the commutativity of the preceding diagram):
G Ω−−−−→ G
Λ
y x
G(f) −−−−→ G×M.
In particular, if we define F to be the composite Lie groupoid morphism G Λ−→ G(f) → G×M ,
then F covers f : Σ → M and, by the commutativity,
F (p) = (Ω(p), α(p)), (3.2)
where α denotes source projection. But as F : G → G×M must respect the target projections,
denoted β, we also have f(β(p)) = β(F (p)). Now (3.2) gives
f(β(p)) = Ω(p) · α(p),
which proves the proposition. ■
Proof of theorem (for Σ simply-connected). That δf1 and δf2 must be isomorphic when
f2 = ϕ ◦ f1, ϕ ∈ Aut(M), is Proposition 3.4. Suppose δf1 ∼= δf2 and assume initially that Σ is
simply-connected (needed in the proof of the lemma below). By definition, there exists l ∈ G
and a Lie algebroid isomorphism λ : A(f2) → A(f1) such that the following diagram commutes:
A(f1)
δf1−−−−→ g
λ
x yAdl
A(f2)
δf2−−−−→ g.
(3.3)
Arbitrarily fixing a point x0 ∈ Σ, (1.3) gives
δf1
(
A(f1)x0
)
= gf1(x0), δf2
(
A(f2)x0
)
= gf2(x0). (3.4)
For i = 1 or 2, let Ωi : Gi → G denote the global form of the monodromy of δfi, as defined
at (1.5). The Lie algebroid of Gi is A(fi) and, by Lie II for Lie groupoids, there is a unique
Lie groupoid isomorphism Λ: G2 → G1 whose derivative is λ. Taking ω = δfi in the preceding
proposition, we obtain
f1(x) = Ω1(p1) · f1(x0), f2(x) = Ω2(p2) · f2(x0), (3.5)
whenever pi ∈ Gi is an arrow from x0 to x. By the commutativity of (3.3), the Lie groupoid
morphisms Ω2 : G2 → G and p2 7→ lΩ1(Λ(p2))l
−1 have the same derivative, namely δf2, so they
must coincide, because G2 is source-connected:
Ω2(p2) = lΩ1(Λ(p2))l
−1, p2 ∈ G2. (3.6)
Since λ, and hence Λ, covers the identity on Σ, p1 ∈ G1 is an arrow from x0 to x if and only if
p2 := Λ(p1) ∈ G2 is an arrow from x0 to x. This fact and (3.6) allow us to rewrite the second
equation in (3.5) as f2(x) = lΩ1(p1)l
−1 · f2(x0). Or, choosing r ∈ G such that
f2(x0) = lr−1 · f1(x0), (3.7)
we have
f1(x) = Ω1(p1) · f1(x0), f2(x) = lΩ1(p1)r
−1 · f1(x0), (3.8)
whenever p1 ∈ G1 is an arrow from x0 to x.
14 A.D. Blaom
Lemma 3.13. r ∈ G lies in the normaliser of Gf1(x0).
Assuming the lemma holds, there exists, by the characterization of symmetries in Proposi-
tion 3.2, an element ϕ ∈ Aut(M) well-defined by ϕ(g · f1(x0)) = lgr−1 · f1(x0). Then (3.8) gives
us f2(x) = ϕ(f1(x)), as required. ■
Proof of lemma. Since we assume the isotropy groups of the action of G on M are weakly
connected, it suffices to show r ∈ NG
(
G◦
f1(x0)
)
. We claim
Ω1
(
G1
x0
)
= G◦
f1(x0)
, (3.9)
Ω2
(
G2
x0
)
= G◦
f2(x0)
. (3.10)
Since Gi is transitive and source-connected (i ∈ {1, 2}) the restriction of the target projection
of Gi to the source-fibre over x0 is a principal Gi
x0
-bundle over Σ. Since we assume Σ is simply-
connected, Gi
x0
is connected, by the long exact homotopy sequence for this principal bundle.
It follows that (3.9) and (3.10) are consequences of their infinitesimal analogues, which already
appear in (3.4) above.
Because Λ: G2 → G1 is a Lie groupoid isomorphism covering the identity, we have
Λ
(
G2
x0
)
= G1
x0
. (3.11)
We now compute
rG◦
f1(x0)
r−1 = l−1G◦
lr·f1(x0)
l = l−1G◦
f2(x0)
l = l−1Ω2
(
G2
x0
)
l = Ω1
(
Λ
(
G2
x0
))
= Ω1
(
G1
x0
)
= G◦
f1(x0)
.
The second and subsequent equalities in this computation follow from equations (3.7), (3.10),
(3.6), (3.11) and (3.9) respectively. ■
Proof of theorem (general case). If δf1 ∼= δf2 but Σ is not simply-connected, then
δ(f1 ◦ π) ∼= δ(f2 ◦ π), where π : Σ̃ → Σ denotes the universal covering map, as it is not diffi-
cult to see. By the result just proven in the simply-connected case, there exists ϕ ∈ Aut(M)
such that f1 ◦ π = ϕ ◦ f2 ◦ π. But as π is surjective, this immediately implies f1 = ϕ ◦ f2. ■
Summary of results
Suppose f is a primitive of a Maurer–Cartan form ω, so that δf(λ(X)) = Adl ω(X), for some
Lie algebroid morphism λ and element l ∈ G. Then it is not hard to show that f ′(x) =
l−1 · f(x) defines a principal primitive of ω. That is, the existence of primitives already implies
the existence of principal primitives. We may therefore summarise the results cited in the
Introduction and our uniqueness result, Corollary 3.10, as follows:
Theorem 3.14 (main theorem). Let M be a homogeneous G-space and ω : A → g an associated
generalized Maurer–Cartan form, where A is a Lie algebroid over some manifold Σ. Then A
is integrable. Furthermore, ω admits a primitive f : Σ → M if and only if it has constant
monodromy Ωm0
x0
: π1(Σ, x0) → M , for some choice of x0 ∈ Σ and m0 ∈ M with x0
ω−→ m0.
Assuming ω is maximal, and the isotropy groups of the action of G on M are weakly connected,
the primitive f is unique up to symmetry.
We reiterate that “symmetry” is to be understood in the sense Definition 3.1.
A Characterisation of Smooth Maps into a Homogeneous Space 15
Acknowledgements
The author is indebted to a referee who contributed the direct proof of integrability in Section 2.
This substantially simplified the proof of the existence theorem appearing in earlier manuscripts.
We thank Yuri Vyatkin, Sean Curry, Andreas Čap, and Rui Fernandes for helpful discussions.
References
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[3] Burstall F.E., Calderbank D.M.J., Conformal submanifold geometry I–III, arXiv:1006.5700.
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[6] Crainic M., Fernandes R.L., Lectures on integrability of Lie brackets, in Lectures on Poisson Geometry,
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https://doi.org/10.3842/SIGMA.2018.062
https://arxiv.org/abs/1703.03851
https://arxiv.org/abs/1006.5700
https://doi.org/10.4007/annals.2003.157.575
https://arxiv.org/abs/math.DG/0105033
https://doi.org/10.2140/gt
https://arxiv.org/abs/math.DG/0611259
https://doi.org/10.1007/b137493
https://doi.org/10.1090/gsm/061
https://doi.org/10.1017/CBO9781107325883
1 Introduction
2 Integrability
3 The uniqueness of primitives up to symmetry
References
|
| id | nasplib_isofts_kiev_ua-123456789-211639 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-20T21:37:01Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Blaom, Anthony D. 2026-01-07T13:43:16Z 2022 A Characterisation of Smooth Maps into a Homogeneous Space. Anthony D. Blaom. SIGMA 18 (2022), 029, 15 pages 1815-0659 2020 Mathematics Subject Classification: 53C99; 22A99; 53D17 arXiv:1702.02717 https://nasplib.isofts.kiev.ua/handle/123456789/211639 https://doi.org/10.3842/SIGMA.2022.029 We generalize Cartan's logarithmic derivative of a smooth map from a manifold into a Lie group to smooth maps into a homogeneous space = /, and determine the global monodromy obstruction to reconstructing such maps from infinitesimal data. The logarithmic derivative of the embedding of a submanifold Σ ⊂ becomes an invariant of Σ under symmetries of the ''Klein geometry'' , whose analysis is taken up in [SIGMA 14 (2018), 062, 36 pages, arXiv:1703.03851]. The author is indebted to a referee who contributed the direct proof of integrability in Section 2. This substantially simplified the proof of the existence theorem appearing in earlier manuscripts. We thank Yuri Vyatkin, Sean Curry, Andreas Čap, and Rui Fernandes for helpful discussions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Characterisation of Smooth Maps into a Homogeneous Space Article published earlier |
| spellingShingle | A Characterisation of Smooth Maps into a Homogeneous Space Blaom, Anthony D. |
| title | A Characterisation of Smooth Maps into a Homogeneous Space |
| title_full | A Characterisation of Smooth Maps into a Homogeneous Space |
| title_fullStr | A Characterisation of Smooth Maps into a Homogeneous Space |
| title_full_unstemmed | A Characterisation of Smooth Maps into a Homogeneous Space |
| title_short | A Characterisation of Smooth Maps into a Homogeneous Space |
| title_sort | characterisation of smooth maps into a homogeneous space |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211639 |
| work_keys_str_mv | AT blaomanthonyd acharacterisationofsmoothmapsintoahomogeneousspace AT blaomanthonyd characterisationofsmoothmapsintoahomogeneousspace |