Difference between revisions of "User:Yakovenko/sandbox1"
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For an Ehresmann connection on a ''vector'' bundle the tangent space to the linear fiber $F\simeq\R^n$ is naturally identified with itself, thus in a local trivialization $\R^n\times U_\alpha$ the connection form can be associated with the tuple of 1-forms $\omega_{i\alpha}=\rd v_i-\sum_{k=1}^m F_{ik\alpha}(v,b)\rd b_k$, where $m=\dim B$, and $i=1,\dots,n=\dim F$. If the connection is affine (see above), then the dependence of $F_{ik\alpha}(v,b)$ on $v$ must be linear. This allows to write the (vector) connection form $\boldsymbol\omega_\alpha$ as | For an Ehresmann connection on a ''vector'' bundle the tangent space to the linear fiber $F\simeq\R^n$ is naturally identified with itself, thus in a local trivialization $\R^n\times U_\alpha$ the connection form can be associated with the tuple of 1-forms $\omega_{i\alpha}=\rd v_i-\sum_{k=1}^m F_{ik\alpha}(v,b)\rd b_k$, where $m=\dim B$, and $i=1,\dots,n=\dim F$. If the connection is affine (see above), then the dependence of $F_{ik\alpha}(v,b)$ on $v$ must be linear. This allows to write the (vector) connection form $\boldsymbol\omega_\alpha$ as | ||
$$ | $$ | ||
− | \omega_\alpha=\rd v_\alpha-\Omega_\alpha\cdot v_\alpha,\qquad\Omega_\alpha=\begin{pmatrix} | + | \boldsymbol\omega_\alpha=\rd v_\alpha-\Omega_\alpha\cdot v_\alpha,\qquad\Omega_\alpha=\begin{pmatrix} |
\theta_{11,\alpha}&\cdots&\theta_{1n,\alpha}\\ | \theta_{11,\alpha}&\cdots&\theta_{1n,\alpha}\\ | ||
\vdots &\ddots&\vdots\\ | \vdots &\ddots&\vdots\\ | ||
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with $\theta_{ij,\alpha}$ being 1-forms on the base $U_\alpha\subseteq B$, $v_\alpha=(v_{1,\alpha},\dots,v_{n,\alpha})$ the coordinates on the fiber and $\rd v_\alpha$ their respective differentials. The matrix 1-form $\Omega_\alpha$ is called the [[matrix connection form]] in the given trivialization. | with $\theta_{ij,\alpha}$ being 1-forms on the base $U_\alpha\subseteq B$, $v_\alpha=(v_{1,\alpha},\dots,v_{n,\alpha})$ the coordinates on the fiber and $\rd v_\alpha$ their respective differentials. The matrix 1-form $\Omega_\alpha$ is called the [[matrix connection form]] in the given trivialization. | ||
− | Passing from one trivialization to another (over $U_\beta$) means replacing the coordinate vector $v_\alpha$ by the new coordinate vector $v_\beta=M_{\beta\alpha}v_\alpha$ with $M_{\alpha\beta}(\cdot)=M^{-1}_{\beta\alpha}(\cdot)$ being a smooth matrix-valued function (cocycle) on $U_{\alpha\beta}$. Applying this gauge transformation, we conclude by the Leibniz rule<ref>Here $\rd M(\cdot)$ is the matrix 1-form on $U_{\alpha\beta}$ whose components are the differentials of the entries of the matrix function $M(\cdot)$.</ref> that | + | Passing from one trivialization to another (over $U_\beta$) means replacing the coordinate vector $v_\alpha$ by the new coordinate vector $v_\beta=M_{\beta\alpha}\cdot v_\alpha$ with $M_{\alpha\beta}(\cdot)=M^{-1}_{\beta\alpha}(\cdot)$ being a smooth matrix-valued function (cocycle) on $U_{\alpha\beta}$. Applying this gauge transformation, we conclude by the Leibniz rule<ref>Here $\rd M(\cdot)$ is the matrix 1-form on $U_{\alpha\beta}$ whose components are the differentials of the entries of the matrix function $M(\cdot)$.</ref> that |
$$ | $$ | ||
− | \rd v_\beta=\rd (M_{\beta\alpha}\cdot v_\alpha)=\rd M_{\beta\alpha}\cdot v_\alpha+M_{\beta\alpha}\cdot \rd v_\alpha=(\rd M_{\beta\alpha}+M_{\beta\alpha}\Omega_\alpha)\cdot M_{\alpha\beta}v_\beta, | + | \rd v_\beta=\rd (M_{\beta\alpha}\cdot v_\alpha)=\rd M_{\beta\alpha}\cdot v_\alpha+M_{\beta\alpha}\cdot \rd v_\alpha=(\rd M_{\beta\alpha}+M_{\beta\alpha}\Omega_\alpha)\cdot M_{\alpha\beta}\cdot v_\beta, |
$$ | $$ | ||
that is, | that is, | ||
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(note the mnemonic order of indices $\alpha,\beta$). The transformation law (GC), sometimes called the ''gauge transformation of the connection matrux form'', involves the "matrix logarithmic derivative" term $\rd M\cdot M^{-1}$ and hence the matrices $\Omega_\alpha$ "do not form a tensor". | (note the mnemonic order of indices $\alpha,\beta$). The transformation law (GC), sometimes called the ''gauge transformation of the connection matrux form'', involves the "matrix logarithmic derivative" term $\rd M\cdot M^{-1}$ and hence the matrices $\Omega_\alpha$ "do not form a tensor". | ||
− | + | ==Covariant derivative== | |
If $s:B\to E$ is a smooth section and $w\in T_b B$ a vector tangent to the base at some point $b\in B$, $\rd s(b)\cdot w\in T_xE$ is a vector tangent to $E$ at $x=s(b)$. The "vertical component" of the this vector, $v_x\cdot \rd s(b)\cdot w\in T_x F_x$ is called the ''covariant derivative'' of the section $s$ in direction of $w$. The usual notation is | If $s:B\to E$ is a smooth section and $w\in T_b B$ a vector tangent to the base at some point $b\in B$, $\rd s(b)\cdot w\in T_xE$ is a vector tangent to $E$ at $x=s(b)$. The "vertical component" of the this vector, $v_x\cdot \rd s(b)\cdot w\in T_x F_x$ is called the ''covariant derivative'' of the section $s$ in direction of $w$. The usual notation is |
Revision as of 13:34, 19 May 2012
Connection
$\def\G{\varGamma}$
An infinitesimal structure on a (smooth) bundle which allows to define the parallel transport between fibres of the bundle.
Historical overview
Initially the notion of connection appeared in the works by Gregorio Ricci and Tullio Levi-Civita who constructed the infinitesimal parallel transport between tangent spaces of a Riemannian manifold. Later the construction was distilled from the Riemannian context by Éli Cartan who constructed connections on principal bundles (equipped by action of a Lie group) and introduced the notion of the connection form. Finally, Charles Ehresmann gave a general definition of connection via distribution of horizontal subspaces, which is the most general construction using only the smooth structure and local triviality of the bundle.
In practice when dealing with connections, one distinguishes several particular cases of bundles and several (interrelated) ways to describe the infinitesimal parallel transport.
- Connections on the vector bundles, defined by an operator of covariant derivation;
- Affine connections on manifolds, defined on the tangent (and cotangent) bundle to a smooth manifold;
- Connections on $G$-bundles equipped with fibrewise action of a Lie group $G$;
- Levi-Civita (metric) connections on Riemannian manifolds, defined on the tangent bundle and preserving the metric structure.
Ehresmann connection
Let $\pi:E\to B$ be a smooth bundle with a generic fiber $F$. The Ehresmann connection is a "horizontal" subbundle $\G\subset TE$ of the tangent bundle, complementary to the "vertical" subbundle $\operatorname{Ker}\rd \pi$. In other words, the tangent space $T_xE$ at any point $x\in E$ is split as a direct sum of the vertical subspace $V_x=T_x F_x$ tangent to the fiber $F_x=\pi^{-1}(\pi(x))$ through $x$, and a complementary subspace $\G_x\subset T_x$; by definition of complementarity, $\dim\G_x=\dim B$, and the restriction $\rd \pi(x)|_{\G_x}\to T_{\pi(x)}B$ is an isomorphism.
This splitting allows to define the parallel transport along any smooth curve $\gamma:[0,1]\to B$ between the fibers $F_0=\pi^{-1}(\gamma(0))$ and $F_1=\pi^{-1}(\gamma(1))$ as follows. Consider the induced bundle $\pi'=\gamma^*\pi$ over the new base $B'=[0,1]$: because of the triviality of the segment $[0,1]$, this bundle is trivial, with the total space $E'\simeq F\times [0,1]$, and the horizontal subbundle $\G$ induces a horizontal subbundle $\G'=\gamma^*\G$ on $E'$. But since the base is now one-dimensional, the distribution $\G'$ is integrable and defines the foliation on $E'$, everywhere transversal to the fibers. Under mild additional conditions[1] the leaf (integral trajectory of the integrable distribution $\G'$) passing through the point $x\in F_0=F\times\{0\}$, crosses the fiber $F_1=F\times\{1\}$ at a unique point $\tau_0^1(x)\in F_1$. By the general theorems from ordinary differential equations, the map $\tau=\tau_0^1$ is a diffeomorphism (smooth and invertible self-map) of $F$.
$G$-invariant connection
Assume that $\pi:E\to B$ is a bundle with a Lie group $G$ acting freely and transitively on the fibers (and hence on the total space), say, by the right multiplication, $E\times G\to E$, $r:(x,g)\mapsto x\cdot g$, $\pi(x\cdot g)=\pi(x)$. Then is natural to consider the equivariant ($G$-invariant) connections for which the horizontal subspaces $\G_x$ and $\G_{x\cdot g}$ are related by the differential of the corresponding right shift $\rd r(\cdot,g)$.
Such invariance means that for any smooth curve $\gamma$ the corresponding parallel transport $\tau_\gamma:F_0\to F_1$ commutes with the action of $G$: $\tau(x\cdot g)=\tau(x)\cdot g$ for any $g\in G$. In other words, it is sufficient to construct the parallel transport for only one point on the fiber.
Note also that for principal $G$-bundles the tangent spaces $T_x F_b$ to the same fiber $\pi^{-1}(b)$ are canonically isomorphic to each other and to the tangent space $T_e G=\mathfrak g$ which is a Lie algebra of the group $G$. The isomorphism is defined by the differential of the map $(G,e)\to (F,x)$, $g\mapsto x\cdot g$.
Affine connection
If the fiber $F$ of a bundle $\pi:E\to B$ is a (finite-dimensional) vector space, then it is natural to consider affine connections. By definition, a connection is affine, if all the parallel transport operators are linear (invertible) maps between the corresponding fibers.
The linear structure on the generic fiber induces the structure of a module (over the ring $C^\infty(B)$ of smooth functions on the base) on the space of sections: for any two sections $s_1,s_2:B\to E$ and any smooth function $f:B\to \R$ the sum $s_1+s_2:B\to E$ and the products $f\cdot s_i:B\to E$ are well-defined smooth sections of $\pi$.
Connection 1-forms (Lie-algebra-valued and matrix-valued)
For each Ehresmann connection on the principal bundle the splitting of the tangent space $T_x E=V_x\oplus \G_x$ into the vertical and horizontal subspaces defines the linear projection of $T_x E$ into the vertical component $V_x$ parallel to the horizontal component $\G_x$. This projection can be interpreted as $\mathfrak g$-valued[2] differential 1-form $\boldsymbol\omega$ on $TE$, which is "identical" on the vertical subspaces (recall that each vertical subspace $V_x$ can be identified with $\mathfrak g$) and invariant by the right action of $G$. The horizontal subspace $\G_x$ by construction is the common null space defined by the Pfaffian equations $\boldsymbol\omega=0$.
If the principal bundle $\pi$ is defined by local trivializations and the corresponding cocycle $\{g_{\alpha\beta}:U_{\alpha\beta}\to G\}$, this means that points from $G\times U_\alpha$ are identified with points from $G\times U_\beta$ over $U_{\alpha\beta}=U_\alpha\cap U_\beta$ by the gauge transformation rule $$ (v_\alpha,b)\simeq (v_\beta,b)\ \iff\ v_\alpha=g_{\alpha\beta}\cdot v_\beta,\ v_\beta=g_{\beta\alpha}\cdot v_\alpha.\tag{GT} $$
In the trivialization over $U_\alpha$ (resp., over $U_\beta$) the connection is defined the tuple of $\mathfrak g$-valued forms $\boldsymbol\omega_\alpha$ (resp., $\boldsymbol\omega_\beta$). To be self-consistent over the intersection $U_{\alpha\beta}$, one tuple should be transformed into the other by the gauge transformation: for any two sections $s_\alpha$ and $s_\beta=g_{\beta\alpha}\cdot s_\alpha$ over $U_{\alpha\beta}$, the linear operators $\boldsymbol\omega_\alpha\circ \rd s_\alpha: T_b B\to \mathfrak g$ and $\boldsymbol\omega_\beta\circ \rd s_\beta:T_b B\to\mathfrak g$ must coincide after computing the differentials[3].
For an Ehresmann connection on a vector bundle the tangent space to the linear fiber $F\simeq\R^n$ is naturally identified with itself, thus in a local trivialization $\R^n\times U_\alpha$ the connection form can be associated with the tuple of 1-forms $\omega_{i\alpha}=\rd v_i-\sum_{k=1}^m F_{ik\alpha}(v,b)\rd b_k$, where $m=\dim B$, and $i=1,\dots,n=\dim F$. If the connection is affine (see above), then the dependence of $F_{ik\alpha}(v,b)$ on $v$ must be linear. This allows to write the (vector) connection form $\boldsymbol\omega_\alpha$ as $$ \boldsymbol\omega_\alpha=\rd v_\alpha-\Omega_\alpha\cdot v_\alpha,\qquad\Omega_\alpha=\begin{pmatrix} \theta_{11,\alpha}&\cdots&\theta_{1n,\alpha}\\ \vdots &\ddots&\vdots\\ \theta_{n1,\alpha}&\cdots&\theta_{nn,\alpha}\end{pmatrix},\quad v=\begin{pmatrix} v_{1,\alpha}\\ \vdots \\ v_{n,\alpha}\end{pmatrix}, \quad \rd v_\alpha=\begin{pmatrix} \rd v_{1,\alpha}\\ \vdots \\ \rd v_{n,\alpha}\end{pmatrix}, $$ with $\theta_{ij,\alpha}$ being 1-forms on the base $U_\alpha\subseteq B$, $v_\alpha=(v_{1,\alpha},\dots,v_{n,\alpha})$ the coordinates on the fiber and $\rd v_\alpha$ their respective differentials. The matrix 1-form $\Omega_\alpha$ is called the matrix connection form in the given trivialization.
Passing from one trivialization to another (over $U_\beta$) means replacing the coordinate vector $v_\alpha$ by the new coordinate vector $v_\beta=M_{\beta\alpha}\cdot v_\alpha$ with $M_{\alpha\beta}(\cdot)=M^{-1}_{\beta\alpha}(\cdot)$ being a smooth matrix-valued function (cocycle) on $U_{\alpha\beta}$. Applying this gauge transformation, we conclude by the Leibniz rule[4] that $$ \rd v_\beta=\rd (M_{\beta\alpha}\cdot v_\alpha)=\rd M_{\beta\alpha}\cdot v_\alpha+M_{\beta\alpha}\cdot \rd v_\alpha=(\rd M_{\beta\alpha}+M_{\beta\alpha}\Omega_\alpha)\cdot M_{\alpha\beta}\cdot v_\beta, $$ that is, $$ \Omega_\beta=\rd M_{\beta\alpha}\cdot M_{\alpha\beta}+M_{\beta\alpha}\cdot\Omega_\alpha \cdot M_{\alpha\beta} \tag{GC} $$ (note the mnemonic order of indices $\alpha,\beta$). The transformation law (GC), sometimes called the gauge transformation of the connection matrux form, involves the "matrix logarithmic derivative" term $\rd M\cdot M^{-1}$ and hence the matrices $\Omega_\alpha$ "do not form a tensor".
Covariant derivative
If $s:B\to E$ is a smooth section and $w\in T_b B$ a vector tangent to the base at some point $b\in B$, $\rd s(b)\cdot w\in T_xE$ is a vector tangent to $E$ at $x=s(b)$. The "vertical component" of the this vector, $v_x\cdot \rd s(b)\cdot w\in T_x F_x$ is called the covariant derivative of the section $s$ in direction of $w$. The usual notation is $$ \nabla_w s(b)\in T_{s(b)}F_b. $$ Note that by construction the covariant derivative is linear with respect to $w$, as a composition of linear maps $\rd s(b):T_b B\to T_{s(b)}E$ and $v_x:T_x\to V_x$, $x=s(b)$. If $s_1,s_2$ are two connections, then the corresponding covariant derivative of their sum $s_1+s_2$ is tangent to the same fiber $\pi^{-1}(b)$ yet at the new point $s_1(b)+s_2(b)$, ditto for the section $f\cdot s$.
Definition. The Ehresmann connection on a vector bundle is called affine, if the covariant derivation operator is additive and satisfies the Leibniz rule[5]: $$ \nabla_w (s_1+s_2)(b)=\nabla_w s_1(b)+\nabla_w s_2(b),\qquad \nabla_w (f\cdot s)(b)= (\rd f\cdot w)\cdot s(b)+f(b)\cdot \nabla_w s(b)\qquad\forall b\in B. $$
...
- ↑ E.g., if the fiber $F$, is compact, but also if it is a finite-dimensional vector- or $G$-space with $\G'$ linear or $G$-invariant.
- ↑ I.e., a tuple of usual, "scalar" 1-forms, of cardinality equal to $\dim\mathfrak g$.
- ↑ The usual way to write the corresponding formulas explicitly involves canonical $\mathfrak g$-valued 1-form on the Lie group, associated with the right shift action of $G$ on itself.
- ↑ Here $\rd M(\cdot)$ is the matrix 1-form on $U_{\alpha\beta}$ whose components are the differentials of the entries of the matrix function $M(\cdot)$.
- ↑ If we formally declare that $\nabla_w f=\rd f\cdot w\in\R^1$, then the Leibniz formula will take the familiar form $\nabla (f\cdot s)=f\cdot (\nabla s)+(\nabla f)\cdot s$ with respect to the linear structure on $F$
Let be a smooth locally trivial fibration with typical fibre
on which a Lie group
acts effectively and smoothly. A connection on this fibre bundle is a mapping of the category of piecewise-smooth curves in the base
into the category of diffeomorphisms of the fibres that associates with a curve
(with initial point
and end point
) a diffeomorphism
satisfying the following axioms:
1) for ,
,
, and
one has
![]() |
2) for an arbitrary trivializing diffeomorphism and for an
, the diffeomorphism
, where
, is defined by the action of some element
;
3) for an arbitrary piecewise-smooth parametrization , the mapping
, where
is the image of
under
, defines a piecewise-smooth curve in
that starts from the unit element
; moreover,
with a common non-zero tangent vector
define paths in
with a common tangent vector
that depends smoothly on
and
.
The diffeomorphism is called the parallel displacement along
. The parallel displacements along all possible closed curves
form the holonomy group of the connection
at
; this group is isomorphic to a Lie subgroup of
that does not depend on
. A curve
on
is said to be horizontal for
if
for any
and some piecewise-smooth parametrization of it. If
and
are given, then there always exists a unique horizontal curve
, called the horizontal lift of the curve
, such that
; it consists of the points
. The set of horizontal lifts of all curves
in
determines the connection
uniquely:
maps the end points of all lifted curves of
into the initial points.
A connection is called linear if depends linearly on
for any
and
, or equivalently, if for any
the tangent vectors of the horizontal curves beginning at
form a vector subspace
of
, called the horizontal subspace. Here
, where
is the fibre through
, that is,
. The smooth distribution
is called the horizontal distribution of the linear connection
. It determines
uniquely: its integral curves are the horizontal lifts.
A fibre bundle is called principal (respectively, a space of homogeneous type), and is denoted by
(respectively,
), if
acts simply transitively (respectively, transitively) on
, that is, if for any
there is exactly one (respectively, there is an) element
that sends
to
. Suppose that
acts on
from the left; then a natural action from the right is defined on
, where
defines
. Here
is identified with the quotient manifold
formed by the orbits
, where
is the stationary subgroup of a point from
. More generally,
can be identified with the quotient manifold
of orbits
relative to the action defined by
.
A smooth distribution on
is a horizontal distribution of some linear connection (which it determines uniquely) if and only if
![]() |
for arbitrary and
. All horizontal distributions on
(respectively,
) are the images of such
under the canonical projection
(respectively, the natural lifts of such
to
under the canonical projection
). Often a linear connection is defined directly as a distribution with the properties mentioned above. It is known that on each
, and so on every
and
, there is a linear connection.
A linear connection in is usually studied by using its connection form, which determines it uniquely and can be the basis for another definition. An important characteristic of a linear connection is the curvature form; this can be used to compute the Lie algebra of the holonomy group.
The idea of a connection first arose in 1917 in the work of T. Levi-Civita [1] on parallel displacement of a vector in Riemannian geometry. The notion of an affine connection was introduced by H. Weyl in 1918. In the 1920s E. Cartan (see [3]–[5]) investigated projective and conformal connections (cf. Projective connection; Conformal connection). In 1926 he gave the general concept of a "non-holonomic space with a fundamental group" (see Connections on a manifold), and identified these spaces from the point of view of the general theory of connections. In the 1940s V.V. Vagner developed an even more general concept that is close in spirit (but not in terms of the method) to the modern idea of a connection. 1950 was a decisive year; in it there appeared the survey by Vagner [6], the first notes of G.F. Laptev, which disclosed new approaches, especially analytic ones, and the work of C. Ehresmann [7] that laid the foundation of the modern global theory of connections. See also Weyl connection; Linear connection; Riemannian connection; Symplectic connection; Hermitian connection.
References
[1] | T. Levi-Civita, "Nozione di parallelismo in una varietà qualunque e consequente specificazione geometrica della curvatura Riemanniana" Rend. Cir. Mat. Palermo , 42 (1917) pp. 173–205 |
[2] | H. Weyl, "Raum, Zeit, Materie" , Springer (1923) |
[3] | E. Cartan, "Les espaces à connexion conforme" Ann. Soc. Polon. Math. , 2 (1924) pp. 171–221 |
[4] | E. Cartan, "Sur les variétés à connexion projective" Bull. Soc. Math. France , 52 (1924) pp. 205–241 |
[5] | E. Cartan, "Les groupes d'holonomie des espaces généralisés" Acta Math. , 48 (1926) pp. 1–42 |
[6] | V.V. Vagner, "Theory of a composite manifold" Trudy Sem. Vektor i Tenzor Anal. , 8 (1950) pp. 11–72 (In Russian) |
[7] | Ch. Ehresmann, "Les connexions infinitésimal dans une espace fibré différentiable" , Colloq. de Topologie Bruxelles, 1950 , G. Thone & Masson (1951) pp. 29–55 |
[8] | G.F. Laptev, "Differential geometry of imbedded manifolds. Group-theoretical method of differential-geometric investigations" Trudy Moskov. Mat. Obshch. , 2 (1953) pp. 275–382 (In Russian) |
[9] | K. Nomizu, "Lie groups and differential geometry" , Math. Soc. Japan (1956) |
[10] | A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1955) (Translated from French) |
[11] | Ü.G. Lumiste, "Connection theory in bundle spaces" J. Soviet Math. , 1 (1973) pp. 363–390 Itogi Nauk. Ser. Algebra. Topol. Geom. 1969 , 21 (1971) pp. 123–168 |
Comments
Consider a smooth locally trivial fibre bundle . A smooth section is a smooth mapping
such that
. This concept generalizes that of a function
(where
is the fibre of
), which is the same as a section of the trivial fibre bundle
. In several areas of mathematics it is important to consider sections instead of just functions. E.g. in gauge field theory. But then one would also like to have something like the partial derivatives of a section available, i.e. the quantity that describes to first order how
changes as
varies (infinitesimally). This requires comparing the fibres of
at neighbouring points, but there is nothing in the concept of a fibre bundle as it stands that allows one to do this. For this some extra structure is needed, and that is provided by the idea of a connection.
It would be simplest if for every two points one could prescribe an isomorphism
in a consistent way, i.e. such that
for all triples
. Here
, the fibre of
over
, is of course
. This, however, would make the bundle trivial, and this is in general not possible. The next best thing would be to have for every smooth path
from
to
an isomorphism
(which may depend on the path
) from the fibre at the initial point of the path to the fibre at the final point, subject to certain natural restrictions. This is precisely what a connection is.
There are — at least — three intuitively natural ways of describing a connection.
i) Provide for every smooth path from
to
an isomorphism
subject to the three conditions 1), 2), 3).
ii) For each let
be the kernel of the tangent mapping at
. The subspace
of the tangent space
to
at
is called the vertical tangent subspace to
at
. Now for each
define a complementary subspace
at
, called the horizontal tangent subspace at
. Thus,
and
induces an isomorphism
. The
are required to vary smoothly with
. In the case of linear connections, cf. above, this is the infinitesimal version of i).
iii) Let be a vector bundle. Then a linear connection can also be specified by giving so to speak the partial derivatives of a section directly (covariant differentiation). This leads to the specification of a bilinear mapping
, where
is the space of vector fields on
and
is the space of sections of
, with certain properties; cf. Linear connection for these properties in the case
. One consequence of these properties is that
,
,
, depends only on
at
. If
is a smooth path starting in
with tangent vector
at
, then
![]() |
where is parallel displacement defined by
.
An elegant and convenient way to describe a linear connection in the case that is a vector bundle is as follows. Let
![]() |
be a local chart of and a trivialization of
. Then above
one has the following local trivialization of
:
![]() |
where the right-hand arrow is projection in the first two factors. A linear connection on is now given by a bundle mapping
(i.e. the diagram
![]() |
is commutative, and is linear in the fibres), such that locally the mapping looks like
![]() |
![]() |
The are the Christoffel symbols (relative to the trivialization
; in case
,
can be taken equal to
so that the Christoffel symbols depend only on the chart
).
Given the connection , the horizontal subspace
is defined by
![]() |
and the covariant derivative of a section along a vector field
is the section
.
In the case of infinite-dimensional manifolds and bundles this last notion of a linear connection appears to be the appropriate replacement of the more traditional covariant derivative , cf. [a2], Sect. 1.1.
References
[a1] | S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963) pp. Chapt. 4 |
[a2] | W. Klingenberg, "Lectures on closed geodesics" , Springer (1979) |
Yakovenko/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Yakovenko/sandbox1&oldid=26713