Frame bundle

In mathematics, a frame bundle is a principal fiber bundle ${\displaystyle F(E)}$ associated with any vector bundle ${\displaystyle E}$. The fiber of ${\displaystyle F(E)}$ over a point ${\displaystyle x}$ is the set of all ordered bases, or frames, for ${\displaystyle E_{x}}$. The general linear group acts naturally on ${\displaystyle F(E)}$ via a change of basis, giving the frame bundle the structure of a principal ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$-bundle (where k is the rank of ${\displaystyle E}$).

Contents

• Definition and construction
• Principal bundle structure
• Associated vector bundles
• Tangent frame bundle
• Smooth frames
• Solder form
• Orthonormal frame bundle
• G-structures
• References

The frame bundle of a smooth manifold is the one associated with its tangent bundle. For this reason it is sometimes called the tangent frame bundle.

Definition and construction

Let ${\displaystyle E\to X}$ be a real vector bundle of rank ${\displaystyle k}$ over a topological space ${\displaystyle X}$. A frame at a point ${\displaystyle x\in X}$ is an ordered basis for the vector space ${\displaystyle E_{x}}$. Equivalently, a frame can be viewed as a linear isomorphism

${\displaystyle p:\mathbf {R} ^{k}\to E_{x}.}$

The set of all frames at ${\displaystyle x}$, denoted ${\displaystyle F_{x}}$, has a natural right action by the general linear group ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$ of invertible ${\displaystyle k\times k}$ matrices: a group element ${\displaystyle g\in \mathrm {GL} (k,\mathbb {R} )}$ acts on the frame ${\displaystyle p}$ via composition to give a new frame

${\displaystyle p\circ g:\mathbf {R} ^{k}\to E_{x}.}$

This action of ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$ on ${\displaystyle F_{x}}$ is both free and transitive (this follows from the standard linear algebra result that there is a unique invertible linear transformation sending one basis onto another). As a topological space, ${\displaystyle F_{x}}$ is homeomorphic to ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$ although it lacks a group structure, since there is no "preferred frame". The space ${\displaystyle F_{x}}$ is said to be a ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$-torsor.

The frame bundle of ${\displaystyle E}$, denoted by ${\displaystyle F(E)}$ or ${\displaystyle F_{\mathrm {GL} }(E)}$, is the disjoint union of all the ${\displaystyle F_{x}}$:

${\displaystyle \mathrm {F} (E)=\coprod _{x\in X}F_{x}.}$

Each point in ${\displaystyle F(E)}$ is a pair (x, p) where ${\displaystyle x}$ is a point in ${\displaystyle X}$ and ${\displaystyle p}$ is a frame at ${\displaystyle x}$. There is a natural projection ${\displaystyle \pi :F(E)\to X}$ which sends ${\displaystyle (x,p)}$ to ${\displaystyle x}$. The group ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$ acts on ${\displaystyle F(E)}$ on the right as above. This action is clearly free and the orbits are just the fibers of ${\displaystyle \pi }$.

Associated vector bundles

A vector bundle ${\displaystyle E}$ and its frame bundle ${\displaystyle F(E)}$ are associated bundles. Each one determines the other. The frame bundle ${\displaystyle F(E)}$ can be constructed from ${\displaystyle E}$ as above, or more abstractly using the fiber bundle construction theorem. With the latter method, ${\displaystyle F(E)}$ is the fiber bundle with same base, structure group, trivializing neighborhoods, and transition functions as ${\displaystyle E}$ but with abstract fiber ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$, where the action of structure group ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$ on the fiber ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$ is that of left multiplication.

Given any linear representation ${\displaystyle \rho$ :\mathrm {GL} (k,\mathbb {R} )\to \mathrm {GL} (V,\mathbb {F} )} there is a vector bundle

${\displaystyle \mathrm {F} (E)\times _{\rho }V}$

associated with ${\displaystyle F(E)}$ which is given by product ${\displaystyle F(E)\times V}$ modulo the equivalence relation ${\displaystyle (pg,v)\sim (p,\rho (g)v)}$ for all ${\displaystyle g}$ in ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$. Denote the equivalence classes by ${\displaystyle [p,v]}$.

The vector bundle ${\displaystyle E}$ is naturally isomorphic to the bundle ${\displaystyle F(E)\times _{\rho }\mathbb {R} ^{k}}$ where ${\displaystyle \rho }$ is the fundamental representation of ${\displaystyle \mathrm {GL} (k,\mathbb {R} )}$ on ${\displaystyle \mathbb {R} ^{k}}$. The isomorphism is given by

${\displaystyle [p,v]\mapsto p(v)}$

where ${\displaystyle v}$ is a vector in ${\displaystyle \mathbb {R} ^{k}}$ and ${\displaystyle p:\mathbb {R} ^{k}\to E_{x}}$ is a frame at ${\displaystyle x}$. One can easily check that this map is well-defined.

Any vector bundle associated with ${\displaystyle E}$ can be given by the above construction. For example, the dual bundle of ${\displaystyle E}$ is given by ${\displaystyle F(E)\times _{\rho ^{*}}(\mathbb {R} ^{k})^{*}}$ where ${\displaystyle \rho ^{*}}$ is the dual of the fundamental representation. Tensor bundles of ${\displaystyle E}$ can be constructed in a similar manner.

Tangent frame bundle

The tangent frame bundle (or simply the frame bundle) of a smooth manifold ${\displaystyle M}$ is the frame bundle associated with the tangent bundle of ${\displaystyle M}$. The frame bundle of ${\displaystyle M}$ is often denoted ${\displaystyle FM}$ or ${\displaystyle \mathrm {GL} (M)}$ rather than ${\displaystyle F(TM)}$. In physics, it is sometimes denoted ${\displaystyle LM}$. If ${\displaystyle M}$ is ${\displaystyle n}$-dimensional then the tangent bundle has rank ${\displaystyle n}$, so the frame bundle of ${\displaystyle M}$ is a principal ${\displaystyle \mathrm {GL} (n,\mathbb {R} )}$ bundle over ${\displaystyle M}$.

Orthonormal frame bundle

If a vector bundle ${\displaystyle E}$ is equipped with a Riemannian bundle metric then each fiber ${\displaystyle E_{x}}$ is not only a vector space but an inner product space. It is then possible to talk about the set of all orthonormal frames for ${\displaystyle E_{x}}$. An orthonormal frame for ${\displaystyle E_{x}}$ is an ordered orthonormal basis for ${\displaystyle E_{x}}$, or, equivalently, a linear isometry

${\displaystyle p:\mathbb {R} ^{k}\to E_{x}}$

where ${\displaystyle \mathbb {R} ^{k}}$ is equipped with the standard Euclidean metric. The orthogonal group ${\displaystyle \mathrm {O} (k)}$ acts freely and transitively on the set of all orthonormal frames via right composition. In other words, the set of all orthonormal frames is a right ${\displaystyle \mathrm {O} (k)}$-torsor.

The orthonormal frame bundle of ${\displaystyle E}$, denoted ${\displaystyle F_{\mathrm {O} }(E)}$, is the set of all orthonormal frames at each point ${\displaystyle x}$ in the base space ${\displaystyle X}$. It can be constructed by a method entirely analogous to that of the ordinary frame bundle. The orthonormal frame bundle of a rank ${\displaystyle k}$ Riemannian vector bundle ${\displaystyle E\to X}$ is a principal ${\displaystyle \mathrm {O} (k)}$-bundle over ${\displaystyle X}$. Again, the construction works just as well in the smooth category.

If the vector bundle ${\displaystyle E}$ is orientable then one can define the oriented orthonormal frame bundle of ${\displaystyle E}$, denoted ${\displaystyle F_{\mathrm {SO} }(E)}$, as the principal ${\displaystyle \mathrm {SO} (k)}$-bundle of all positively oriented orthonormal frames.

If ${\displaystyle M}$ is an ${\displaystyle n}$-dimensional Riemannian manifold, then the orthonormal frame bundle of ${\displaystyle M}$, denoted ${\displaystyle F_{\mathrm {O} }(M)}$ or ${\displaystyle \mathrm {O} (M)}$, is the orthonormal frame bundle associated with the tangent bundle of ${\displaystyle M}$ (which is equipped with a Riemannian metric by definition). If ${\displaystyle M}$ is orientable, then one also has the oriented orthonormal frame bundle ${\displaystyle F_{\mathrm {SO} }M}$.

Given a Riemannian vector bundle ${\displaystyle E}$, the orthonormal frame bundle is a principal ${\displaystyle \mathrm {O} (k)}$-subbundle of the general linear frame bundle. In other words, the inclusion map

${\displaystyle i:{\mathrm {F} }_{\mathrm {O} }(E)\to {\mathrm {F} }_{\mathrm {GL} }(E)}$

is principal bundle map. One says that ${\displaystyle F_{\mathrm {O} }(E)}$ is a reduction of the structure group of ${\displaystyle F_{\mathrm {GL} }(E)}$ from ${\displaystyle \mathrm {GL} (n,\mathbb {R} )}$ to ${\displaystyle \mathrm {O} (k)}$.

G-structures

If a smooth manifold ${\displaystyle M}$ comes with additional structure it is often natural to consider a subbundle of the full frame bundle of ${\displaystyle M}$ which is adapted to the given structure. For example, if ${\displaystyle M}$ is a Riemannian manifold we saw above that it is natural to consider the orthonormal frame bundle of ${\displaystyle M}$. The orthonormal frame bundle is just a reduction of the structure group of ${\displaystyle F_{\mathrm {GL} }(M)}$ to the orthogonal group ${\displaystyle \mathrm {O} (n)}$.

In general, if ${\displaystyle M}$ is a smooth ${\displaystyle n}$-manifold and ${\displaystyle G}$ is a Lie subgroup of ${\displaystyle \mathrm {GL} (n,\mathbb {R} )}$ we define a G-structure on ${\displaystyle M}$ to be a reduction of the structure group of ${\displaystyle F_{\mathrm {GL} }(M)}$ to ${\displaystyle G}$. Explicitly, this is a principal ${\displaystyle G}$-bundle ${\displaystyle F_{G}(M)}$ over ${\displaystyle M}$ together with a ${\displaystyle G}$-equivariant bundle map

${\displaystyle {\mathrm {F} }_{G}(M)\to {\mathrm {F} }_{\mathrm {GL} }(M)}$

over ${\displaystyle M}$.

In this language, a Riemannian metric on ${\displaystyle M}$ gives rise to an ${\displaystyle \mathrm {O} (n)}$-structure on ${\displaystyle M}$. The following are some other examples.

• Every oriented manifold has an oriented frame bundle which is just a ${\displaystyle \mathrm {GL} ^{+}(n,\mathbb {R} )}$-structure on ${\displaystyle M}$.
• A volume form on ${\displaystyle M}$ determines a ${\displaystyle \mathrm {SL} (n,\mathbb {R} )}$-structure on ${\displaystyle M}$.
• A ${\displaystyle 2n}$-dimensional symplectic manifold has a natural ${\displaystyle \mathrm {Sp} (2n,\mathbb {R} )}$-structure.
• A ${\displaystyle 2n}$-dimensional complex or almost complex manifold has a natural ${\displaystyle \mathrm {GL} (n,\mathbb {C} )}$-structure.

In many of these instances, a ${\displaystyle G}$-structure on ${\displaystyle M}$ uniquely determines the corresponding structure on ${\displaystyle M}$. For example, a ${\displaystyle \mathrm {SL} (n,\mathbb {R} )}$-structure on ${\displaystyle M}$ determines a volume form on ${\displaystyle M}$. However, in some cases, such as for symplectic and complex manifolds, an added integrability condition is needed. A ${\displaystyle \mathrm {Sp} (2n,\mathbb {R} )}$-structure on ${\displaystyle M}$ uniquely determines a nondegenerate 2-form on ${\displaystyle M}$, but for ${\displaystyle M}$ to be symplectic, this 2-form must also be closed.

Related Research Articles

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References

• Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry , vol. 1 (New ed.), Wiley Interscience, ISBN   0-471-15733-3
• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on 2017-03-30, retrieved 2008-08-02
• Sternberg, S. (1983), Lectures on Differential Geometry ((2nd ed.) ed.), New York: Chelsea Publishing Co., ISBN   0-8218-1385-4