G-structure

on a manifold

A principal subbundle with structure group $G$ of the principal bundle of co-frames on the manifold. More exactly, let $\pi _ {k} : M _ {k} \rightarrow M$ be the principal $\mathop{\rm GL} ^ {k} ( n)$- bundle of all co-frames of order $k$ over an $n$- dimensional manifold $M$, and let $G$ be a subgroup of the general linear group $\mathop{\rm GL} ^ {k} ( n)$ of order $k$. A submanifold $P$ of the manifold of $k$- co-frames $M _ {k}$ defines a $G$- structure of order $k$, $\pi = \pi _ {k} \mid _ {P} : P \rightarrow M$, if $\pi$ defines a principal $G$- bundle, i.e. the fibres of $\pi$ are orbits of $G$. For example, a section $x \mapsto u _ {x} ^ {k}$ of $\pi _ {k}$( a field of co-frames) defines a $G$- structure $P = \{ {gu _ {x} ^ {k} } : {x \in M, g \in G } \}$, which is called the $G$- structure generated by this field of co-frames. Any $G$- structure is locally generated by a field of co-frames. The $G$- structure over the space $V = \mathbf R ^ {n}$ generated by the field of co-frames $x \mapsto j _ {x} ^ {k} ( \mathop{\rm id} )$, where $\mathop{\rm id} : V \rightarrow V$ is the identity mapping, is called the standard flat $G$- structure.

Let $\pi : P \rightarrow M$ be a $G$- structure. The mapping of the manifold $P$ into the point $eG \in \mathop{\rm GL} ^ {k} ( n)/G$ can be extended to a $\mathop{\rm GL} ^ {k} ( n)$- equivariant mapping $S: M _ {k} \rightarrow \mathop{\rm GL} ^ {k} ( n)/G$, which can be considered as a structure of type $\mathop{\rm GL} ^ {k} ( n)/G$ on $M$. If the homogeneous space $\mathop{\rm GL} ^ {k} ( n)/G$ is imbedded as an orbit in a vector space $W$ admitting a linear action of $\mathop{\rm GL} ^ {k} ( n)$, then the structure $S$ can be considered as a linear structure of type $W$; this is called the Bernard tensor of the $G$- structure $\pi$, and is often identified with it. Conversely, let $S: M _ {k} \rightarrow W$ be a linear geometric structure of type $W$( for example, a tensor field), whereby $S( M _ {k} )$ belongs to a single orbit $\mathop{\rm GL} ^ {k} ( n) w _ {0}$ of $\mathop{\rm GL} ^ {k} ( n)$. $P = S ^ {-} 1 ( w _ {0} )$ is then a $G$- structure, where $G$ is the stabilizer of the point $w _ {0}$ in $\mathop{\rm GL} ^ {k} ( n)$, and $S$ is its Bernard tensor. For example, a Riemannian metric defines an $O( n)$- structure, an almost-symplectic structure defines a $\mathop{\rm Sp} ( n/2, \mathbf R )$- structure, an almost-complex structure defines a $\mathop{\rm GL} ( n/2, \mathbf C )$- structure, and a torsion-free connection defines a $\mathop{\rm GL} ( n)$- structure of the second order ( $\mathop{\rm GL} ( n)$ is considered here as a subgroup of the group $\mathop{\rm GL} ^ {2} ( n)$). An affinor (a field of endomorphisms) defines a $G$- structure if and only if it has at all points one and the same Jordan normal form $A$, where $G$ is the centralizer of the matrix $A$ in $\mathop{\rm GL} ( n)$.

The elements of the manifold $M _ {k}$ can be considered as co-frames of order 1 on $M _ {k-} 1$, which makes it possible to consider the natural bundle $\pi ^ {k} : M _ {k} \rightarrow M _ {k-} 1$ as an $N ^ {k}$- structure of order one, where $N ^ {k}$ is the kernel of the natural homomorphism $\mathop{\rm GL} ^ {k} ( n) \rightarrow \mathop{\rm GL} ^ {k-} 1 ( n)$. Every $G$- structure $\pi : P \rightarrow M$ of order $k$ has a related sequence of $G$- structures of order one,

$$P \rightarrow P _ {-} 1 \rightarrow P _ {-} 2 \rightarrow \dots \rightarrow P _ {-} k = M,$$

where $P _ {-} i = \pi ^ {k} ( P _ {-} i+ 1 ) \subset M _ {k-} i$. Consequently, the study of $G$- structures of higher order reduces to the study of $G$- structures of order one. A co-frame $u _ {x} ^ {1} \in M _ {1}$ can be considered as an isomorphism $u _ {x} ^ {1} : T _ {x} M \rightarrow V$.

The $1$- form $\theta : TM _ {1} \rightarrow V$, assigning to a vector $X \in T _ {u _ {x} ^ {1} } M _ {1}$ the value $\theta _ {u _ {x} ^ {1} } ( X) = u _ {x} ^ {1} ( \pi _ {1} ) _ \star X$, is called the displacement form. In the local coordinates $( x ^ {i} , u _ {i} ^ {a} )$ of $M _ {1}$, the form $\theta$ is expressed as $\theta = u _ {i} ^ {a} dx ^ {i} \otimes e _ {a}$, where $e _ {a}$ is the standard basis in $V$.

The restriction $\theta _ {P}$ of $\theta$ on a $G$- structure $P \subset M _ {1}$ is called the displacement form of the $G$- structure. It possesses the following properties: 1) strong horizontality: $\theta _ {P} ( X) = 0 \iff \pi _ \star X = 0$; and 2) $G$- equivariance: $\theta _ {P} \circ g = g \circ \theta _ {P}$ for any $g \in G$.

Using the form $\theta _ {P}$ it is possible to characterize the principal bundles with base $M$ that are isomorphic to a $G$- structure. Namely, a principal $G$- bundle $\pi : P \rightarrow M$ is isomorphic to a $G$- structure if and only if there are a faithful linear representation $\alpha$ of the group $G$ in an $n$- dimensional vector space $V$, $n = \mathop{\rm dim} M$, and a $V$- valued strongly-horizontal $G$- equivariant $1$- form $\theta$ on $P$. Removal of the requirement that the representation $\alpha$ be faithful gives the concept of a generalized $G$- structure (of order one) on $M$, namely a principal $G$- bundle $P \rightarrow M$ with a linear representation $\alpha : G \rightarrow \mathop{\rm GL} ( V)$, $\mathop{\rm dim} V = \mathop{\rm dim} M$, and a $V$- valued strongly-horizontal $G$- equivariant $1$- form $\theta$ on $P$.

An example of a generalized $G$- structure is the canonical bundle $\pi : P \rightarrow G \setminus P$ over the homogeneous space $G \setminus P$ of a Lie group $P$. Here $\alpha$ is the isotropy representation of the group $G$, while $\theta$ is defined by the Maurer–Cartan form of $P$.

Let $\pi : P \rightarrow M$ be a $G$- structure of order one. The bundle $\pi ^ \prime : P ^ { \prime } \rightarrow P$ of $1$- jets of local sections of $\pi$ can be considered as a $G ^ { \prime }$- structure on $P$, where $G ^ { \prime } = \mathop{\rm Hom} ( V, \mathfrak g )$ is a commutative group, $\mathfrak g$ is the Lie algebra of $G$, that is linearly represented in the space $V \oplus \mathfrak g$ by the formula

$$A( v, X) = \ ( v, X+ A( v)),\ \ A \in G ^ { \prime } ,\ \ v \in V,\ \ X \in \mathfrak g ,$$

and that acts on the manifold $P ^ { \prime }$ according to the formula

$$H \mapsto AH = \{ {l _ {p} A( \theta ( h)) + h } : {A \in G ^ { \prime } , p = \pi ^ { \prime } ( H) , h \in H } \} ,$$

where $l _ {p}$ is the canonical isomorphism of the Lie algebra $\mathfrak g$ of the group $G$ onto the vertical subspace $T _ {p} ^ {V} P = T _ {p} ( \pi ^ {-} 1 ( \pi ( p)))$. Here the element $H \in P ^ { \prime }$ is considered as a horizontal (i.e. complementary to the vertical) subspace in $T _ {p} P$. It defines a co-frame $\theta _ {H} ^ \prime : T _ {p} P \mathop \rightarrow \limits ^ \approx \mathfrak g + V$, which is defined on a vertical subspace by the mapping $l _ {p}$, and on a horizontal subspace by the mapping $\theta _ {H} = \theta \mid _ {H}$. The vector function $C ^ { \prime } : P ^ { \prime } \rightarrow W = \mathop{\rm Hom} ( V \wedge V, V)$, defined by the formula $H \mapsto C _ {H} ^ { \prime }$, $C _ {H} ^ { \prime } ( u, v) = d \theta ( \theta _ {H} ^ {-} 1 u , \theta _ {H} ^ {-} 1 v)$, is called the torsion function of the $G$- structure $\pi$. A section $s: x \mapsto H _ {p(} x)$ of the bundle $\pi \circ \pi ^ \prime : P ^ { \prime } \rightarrow M$ defines a connection on $\pi$, while the restriction of the function $C ^ { \prime }$ on $s( M)$ is a function defining the coordinates of the torsion tensor of this connection relative to the field of co-frames $p( x)$.

The mapping $C ^ { \prime } : P ^ { \prime } \rightarrow W$ is $G ^ { \prime }$- equivariant relative to the above-mentioned action of $G ^ { \prime }$ on $P$ and to the action of $G ^ { \prime }$ on $W$, which is defined by the formula

$$A: w \mapsto Aw = w + \delta A,$$

where $\delta : G ^ { \prime } \rightarrow W$, $( \delta A)( u, v) = A( u) v - A( v) u$. The mapping $C: P \rightarrow G ^ { \prime } \setminus W$ induced by the mapping $C ^ { \prime }$ is called the structure function of the $G$- structure $\pi$, the vanishing of $C$ is equivalent to the existence of a torsion-free connection on $\pi$.

The choice of a subspace $D \subset W$ complementary to $\delta G ^ { \prime }$ defines a subbundle $P ^ {(} 1) = C ^ { \prime - 1 } ( D)$ of the bundle of co-frames $\pi ^ { \prime } : P ^ { \prime } \rightarrow P$ with structure group $G ^ {(} 1) = G ^ { \prime } \cap \mathop{\rm Ker} \delta \cong \mathfrak g \otimes V ^ \star \cap V \otimes S ^ {2} V ^ \star \subset V \otimes V ^ \star 2$, i.e. a $G ^ {(} 1)$- structure $\pi ^ {(} 1) = \pi ^ \prime \mid _ {P ^ {(} 1) } : P ^ {(} 1) \rightarrow P$ on $P$. It is called the first prolongation of the $G$- structure $\pi$. The $i$- th prolongation $\pi ^ {(} i) : P ^ {(} i) \rightarrow P ^ {(} i- 1)$ is defined by induction as the $G ^ {(} i)$- structure on $P ^ {(} i- 1)$, where the group $G ^ {(} i)$ is isomorphic to the vector group $\mathfrak g \otimes S ^ {i} V ^ \star \cap V \otimes S ^ {i+} 1 V ^ \star \subset V \otimes V ^ {\star(} i+ 1)$. The structure function $C ^ {(} i)$ of the $i$- th prolongation is called the structure function of $i$- th order of the $G$- structure $\pi$.

The central problem of the theory of $G$- structures is the local equivalence problem, i.e. the problem of finding necessary and sufficient conditions under which two $G$- structures $\pi : P \rightarrow M$ and $\overline \pi \; : \overline{P}\; \rightarrow \overline{M}\;$ with the same structure group $G$ are locally equivalent, i.e. a local diffeomorphism $\phi : M \supset U \rightarrow \overline{U}\; \subset \overline{M}\;$ of the manifolds $M$ and $\overline{M}\;$ should exist that induces an isomorphism of $G$- structures over the neighbourhoods $U$ and $\overline{U}\;$. A particular case of this problem is the integrability problem, i.e. the problem of finding necessary and sufficient conditions for the local equivalence of a given $G$- structure and the standard flat $G$- structure. The local equivalence problem can be reformulated as the problem of finding a complete system of local invariants of a $G$- structure.

For an $O( n)$- structure, which is identified with a Riemannian metric, the integrability problem was solved by B. Riemann: Necessary and sufficient conditions for integrability consist in the vanishing of the curvature tensor of the metric. The local equivalence problem was solved by E. Christoffel and R. Lipschitz: A complete system of local invariants of a Riemannian metric consists of its curvature tensor and its successive covariant derivatives (see [1]).

An approach to solving the equivalence problem is based on the concepts of a prolongation and a structure function. Every $G$- structure $\pi : P \rightarrow M$ of order one with structure group $G \subset \mathop{\rm GL} ( n)$ is connected with a sequence of prolongations

$$\dots \rightarrow P ^ {(} i) \rightarrow P ^ {(} i- 1) \rightarrow \dots \rightarrow P \mathop \rightarrow \limits ^ \pi M,$$

and a sequence of structure functions $C ^ {(} i)$. For an $O( n)$- structure, the structure function $C ^ {(} 0) = C$ on $P ^ {(} 0) = P$ is equal to 0, while the essential parts of the remaining structure functions $C ^ {(} i)$, $i > 0$, are identified with the curvature tensor of the corresponding metric and its successive covariant derivatives. For $\pi$ to be integrable it is necessary and sufficient that the structure functions $C ^ {(} 0) \dots C ^ {(} k)$ be constant, and that their values coincide with the corresponding values of the structure functions of the standard flat $G$- structure (see [6], [8], [9]). The number $k$ depends only on the group $G$. For a broad class of linear groups, especially for all irreducible groups $G \subset \mathop{\rm GL} ( n)$ that do not belong to Berger's list of holonomy groups of spaces with a torsion-free affine connection [3], one has $k= 0$, and for a $G$- structure to be integrable it is necessary and sufficient that the structure function $C ^ {(} 0)$ vanishes, or that a torsion-free linear connection exists, preserving the $G$- structure.

A $G$- structure $\pi$ is called a $G$- structure of finite type (equal to $k$) if $G ^ {(} k- 1) \neq \{ e \}$, $G ^ {(} k) = \{ e \}$. In this case $\pi ^ {(} k) : P ^ {(} k) \rightarrow P ^ {(} k- 1)$ is a field of co-frames (an absolute parallelism), and the automorphism group of the $G$- structure $\pi$ is isomorphic to the automorphism group of this parallelism and is a Lie group. The local equivalence problem of these structures reduces to the equivalence problem of absolute parallelisms and has been solved in terms of a finite sequence of structure functions (see [2]). For a $G$- structure of infinite type, the local equivalence problem remains unsolved in the general case (1984).

Two $G$- structures $\pi : P \rightarrow M$ and $\pi ^ \prime : P ^ { \prime } \rightarrow M ^ { \prime }$ are called formally equivalent at the points $x \in M$, $x ^ \prime \in M ^ { \prime }$ if an isomorphism of the fibres $\pi ^ {-} 1 ( x) \rightarrow \pi ^ {-} 1 ( x ^ \prime )$ exists that can be continued to an isomorphism of the corresponding fibres of the prolongations $P ^ {(} i) \rightarrow M$ and $P ^ { \prime ( i) } \rightarrow M ^ { \prime }$ $( i \geq 0)$. Examples have been found which demonstrate that if two $G$- structures of class $C ^ \infty$ are formally equivalent for all pairs $( x, x ^ \prime ) \in M \times M ^ { \prime }$, then it does not follow, generally speaking, that they are locally equivalent [6]. In the analytic case, proper subsets $S( M) \subset M$, $S( M ^ { \prime } ) \subset M ^ { \prime }$ exist, which are countable unions of analytic sets, such that for any $x \in M \setminus S( M)$, $x ^ \prime \in M ^ { \prime } \setminus S( M ^ { \prime } )$, the formal equivalence of two structures $P$ and $P ^ { \prime }$ at the points $x, x ^ \prime$ implies that they are locally equivalent [7].

References

[1]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Wiley (1963)
[2]	S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)
[3]	M. Berger, "Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes" Bull. Soc. Math. France , 83 (1955) pp. 279–330
[4]	S.S. Chern, "The geometry of -structures" Bull. Amer. Math. Soc. , 72 (1966) pp. 167–219
[5]	S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972)
[6]	P. Molino, "Théorie des -structures: le problème d'Aeequivalence" , Springer (1977)
[7]	T. Morimoto, "Sur le problème d'équivalence des structures géométriques" C.R. Acad. Sci. Paris , 292 : 1 (1981) pp. 63–66 (English summary)
[8]	I.M. Singer, S. Sternberg, "The infinite groups of Lie and Cartan. I. The transitive groups" J. d'Anal. Math. , 15 (1965) pp. 1–114
[9]	A.S. Pollack, "The integrability problem for pseudogroup structures" J. Diff. Geom. , 9 : 3 (1974) pp. 355–390