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In [[mathematics]], particularly [[differential topology]], the '''secondary vector bundle structure'''
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refers to the natural [[vector bundle]] structure (''TE'',''p''<sub>*</sub>,''TM'') on the total space ''TE'' of the [[tangent bundle]] of a smooth vector bundle (''E'',''p'',''M''), induced by the push-forward ''p''<sub>*</sub>:''TE''&rarr;''TM'' of the original projection map ''p'':''E''&rarr;''M''.
 
In the special case (''E'',''p'',''M'')=(''TM'',&pi;<sub>''TM''</sub>,''M''), where ''TE''=''TTM'' is the [[double tangent bundle]], the secondary vector bundle (''TTM'',(&pi;<sub>''TM''</sub>)<sub>*</sub>,''TM'') is isomorphic to the [[tangent bundle]]
(''TTM'',&pi;<sub>''TTM''</sub>,''TM'') of ''TM'' through the [[double tangent bundle|canonical flip]].
 
== Construction of the secondary vector bundle structure ==
 
Let (''E'',''p'',''M'') be a smooth vector bundle of rank ''N''. Then the preimage (''p''<sub>*</sub>)<sup>-1</sup>(X)⊂''TE'' of any tangent vector ''X''&isin;''TM'' in the push-forward ''p''<sub>*</sub>:''TE''&rarr;''TM'' of the canonical projection ''p'':''E''&rarr;''M'' is a smooth submanifold of dimension 2''N'', and it becomes a vector space with the push-forwards
:<math>
+_*:T(E\times E)\to TE \quad , \quad \lambda_*:TE\to TE
</math>
of the original addition and scalar multiplication
:<math>
+:E\times E\to E \qquad , \qquad \lambda:E\to E</math>
as its vector space operations. The triple (''TE'',''p''<sub>*</sub>,''TM'') becomes a smooth vector bundle with these vector space operations on its fibres.
 
=== Proof ===
 
Let (''U'',&phi;) be a local coordinate system on the base manifold ''M'' with &phi;(''x'')=(''x''<sup>1</sup>,...,''x''<sup>n</sup>) and let
:<math>
\psi:W \to \varphi(U)\times \mathbb R^N \quad ; \quad \psi(v^k e_k|_x) := (x^1,\ldots,x^n,v^1,\ldots,v^N)
</math>
be a coordinate system on ''E'' adapted to it. Then
:<math>
p_*\Big(X^k\frac{\partial}{\partial x^k}\Big|_v + Y^\ell\frac{\partial}{\partial v^\ell}\Big|_v\Big) = X^k\frac{\partial}{\partial x^k}\Big|_{p(v)},
</math>
so the fiber of the secondary vector bundle structure at ''X''&isin;''T''<sub>''x''</sub>''M'' is of the form
:<math>
p^{-1}_*(X) = \Big\{ \ X^k\frac{\partial}{\partial x^k}\Big|_v + Y^\ell\frac{\partial}{\partial v^\ell}\Big|_v
\ \Big| \ v\in E_x \ , \ Y^1,\ldots,Y^N\in\R \ \Big\}.
</math>
Now it turns out that
:<math>
\chi\Big(X^k\frac{\partial}{\partial x^k}\Big|_v + Y^\ell\frac{\partial}{\partial v^\ell}\Big|_v\Big) = \Big(X^k\frac{\partial}{\partial x^k}\Big|_{p(v)}, (v^1,\ldots,v^N,Y^1,\ldots,Y^N) \Big)
</math>
gives a local trivialization &chi;:''TW''&rarr;''TU''×'''R'''<sup>2''N''</sup> for (''TE'',''p''<sub>*</sub>,''TM''), and the push-forwards of the original vector space operations read in the adapted coordinates as
:<math>
\Big(X^k\frac{\partial}{\partial x^k}\Big|_v + Y^\ell\frac{\partial}{\partial v^\ell}\Big|_v\Big)
 
+_*
\Big(X^k\frac{\partial}{\partial x^k}\Big|_w + Z^\ell\frac{\partial}{\partial v^\ell}\Big|_w\Big)
 
=
X^k\frac{\partial}{\partial x^k}\Big|_{v+w} + (Y^\ell+Z^\ell)\frac{\partial}{\partial v^\ell}\Big|_{v+w}
</math>
and
:<math>
\lambda_*\Big(X^k\frac{\partial}{\partial x^k}\Big|_v + Y^\ell\frac{\partial}{\partial v^\ell}\Big|_v\Big)
 
=
X^k\frac{\partial}{\partial x^k}\Big|_{\lambda v} + \lambda Y^\ell\frac{\partial}{\partial v^\ell}\Big|_{\lambda v},
</math>
so each fibre (''p''<sub>*</sub>)<sup>-1</sup>(''X'')⊂''TE'' is a vector space and the triple (''TE'',''p''<sub>*</sub>,''TM'') is a smooth vector bundle.
 
== Linearity of connections on vector bundles ==
 
The general [[Ehresmann connection]]
:<math>
TE = HE \oplus VE
</math>
on a vector bundle (''E'',''p'',''M'') can be characterized in terms of the '''connector map'''
:<math>
\kappa:T_vE\to E_{p(v)} \qquad ; \qquad \kappa(X):=\operatorname{vl}_v^{-1}(\operatorname{vpr}X),
</math>
where vl<sub>''v''</sub>:''E''&rarr;''V''<sub>''v''</sub>''E'' is the [[vector bundle|vertical lift]], and vpr<sub>''v''</sub>:''T''<sub>''v''</sub>''E''&rarr;''V''<sub>''v''</sub>''E'' is the [[Ehresmann connection|vertical projection]]. The mapping
:<math>
\nabla:TM\times\Gamma(E)\to\Gamma(E) \quad ; \quad \nabla_Xv := \kappa(v_*X)
</math>
induced by an Ehresmann connection is a [[covariant derivative]] on &Gamma;(''E'') in the sense that
*<math> \nabla_{X+Y}v = \nabla_X v + \nabla_Y v</math>
*<math> \nabla_{\lambda X}v=\lambda \nabla_Xv</math>
*<math> \nabla_X(v+w) = \nabla_X v + \nabla_X w</math>
*<math> \nabla_X(\lambda v)=\lambda \nabla_Xv</math>
*<math> \nabla_X(fv) = X[f]v + f\nabla_Xv</math>
if and only if the connector map is linear with respect to the secondary vector bundle structure (''TE'',''p''<sub>*</sub>,''TM'') on ''TE''. Then the connection is called ''linear''. Note that the connector map is automatically linear with respect to the tangent bundle structure (''TE'',&pi;<sub>''TE''</sub>,''E'').
 
== See also ==
* [[Connection (vector bundle)]]
* [[Double tangent bundle]]
* [[Ehresmann connection]]
* [[Vector bundle]]
 
== References ==
 
* P.Michor. ''Topics in Differential Geometry,'' American Mathematical Society (2008).
 
[[Category:Differential geometry]]
[[Category:Topology]]
[[Category:Differential topology]]

Latest revision as of 13:11, 5 May 2014

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If you are using linear pull brakes with drop levers, you must use a lever that is designed to function with the increased mechanical advantage that linear pull brakes afford. If you're planning on doing mostly road biking with the occasional cross country trip, and only once in a while going down an actual mountain. A wider tire will give you more traction when riding over rocks or in turns, you can run at lower pressure and the more cushioned the ride will feel, but they are heavier. Now please continue on additionally so you can get added info for this topic. Now the most successful business gurus are looking to the interns for lessons.

There are aspects of mountain biking that are more similar to trail running than regular bicycling. Equally expensive can be the tag attached to a lighter-weighted bike. The tape (or string against the ruler) will indicate what your helmet size is. Once you reach the top, you will be rewarded with quite a few scenic overlooks and enough short climbs and descents to keep it interesting. You have to make sure that your bike is set as to provide you with maximum comfort you need for a long and rough ride.

Consequently, they became like villains in the eyes of the NBA fans outside Miami. Cold and long winters at high altitude are perfect conditions for winter recreation. If the metal tubing below the paint has become exposed, then touch this up with a dab of enamel paint, using a very fine brush. " I want us to be able to chase one another around the room, have pillow fights, and wrestle. The best way to narrow down your options is to determine the components that are most important to you, such as the forks, rear derailleur and wheels.

Looked like one or two other people beat me to the punch. They get used on the street which is not what it's built for. There are good bikes available at reasonable prices, and there are some overpriced stinkers. Suspension: Road bikes are built with a sole purpose of providing greater speed; they do not possess this feature, although they have certain materials which absorb the shocks of the uneven roads. They will provide you with a honest price, yet be ready to pay in between $650 and $4,000 on dual suspensions and in between $470 and $670 for hardtails.

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