\nI won’t describe general double vector bundles here, all we really need is the tangent bundle of a vector bundle \\(TE \\), which we equip with natural local coordinates \\((x^{A}, y^{a}, \\dot{x}^{A}, \\dot{y}^{a}) \\). The coordinate transformations here are given by<\/p>\n
\\(x^{A’} = x^{A’}(x)\\)
\n\\(y^{a’} = y^{b}T_{b}^{\\:\\: a’}(x)\\)
\n\\(\\dot{x}^{A’} = \\dot{x}^{B}\\frac{\\partial x^{A’}}{\\partial x^{B}}(x)\\)
\n\\(\\dot{y}^{a’} = \\dot{y}^{b}T_{b}^{\\:\\: a’}(x) + y^{b}\\dot{x}^{B}\\frac{\\partial T_{b}^{\\:\\:a’}}{\\partial x^{B}}(x)\\)<\/p>\n
It is now convenient to view this double vector bundle as a bi-graded manifold by assigning a bi-weight as \\(w(x) = (0,0)\\), \\(w(y) = (1,0)\\), \\(w(\\dot{x}) = (0,1)\\) and \\(w(\\dot{y}) = (1,1)\\).<\/p>\n
Note that as a graded manifold the coordinate transformations must respect this bi-grading. A quick look at the transformation laws show that they do preserve this bi-grading. In fact, under some mild conditions, we can always view a double vector bundle as such a bi-graded manifold. This is far more convenient than the original definitions as a vector bundle in the category of vector bundles, where one has to check all the compatibility conditions.<\/p>\n
We have the following vector bundle structures<\/p>\n
![\"DVD\"](\"http:\/\/blogs-new.scienceforums.net\/ajb\/wp-content\/uploads\/sites\/10\/2014\/04\/DVB-1-300x181.png\")
<\/p>\n
The core of this double vector bundle is identified with \\(ker(T\\pi)\\) which is the vertical bundle \\(VE\\), which you can easily see is isomorphic to \\(E\\).<\/p>\n
As we have a bi-graded structure here we can also pass to a graded structure by considering the total weight. That is just sum the components of the bi-weight. In doing so we get what is know as a graded bundle of degree two [2]. In simpler language we have the series of fibrations<\/p>\n
\\(TE \\rightarrow E \\times_{M} TM \\rightarrow M \\),<\/p>\n
defined by first projecting out \\(\\dot{y}\\), which has weight 2 and the then projecting out \\(y\\) and \\(\\dot{x}\\) which have weight 1. Again you can see that this is all consistent with the coordinate transformations.<\/p>\n
Also note that \\(E \\times TM\\) also carries the structure of a double vector bundle simply by projection onto the two factors and then the obvious projections to \\(M\\). This can be viewed keeping the bi-weight as inherited by that on \\(TE \\).<\/p>\n
Connection in a vector bundle<\/strong> Def<\/strong> A (linear) connection on a vector bundle \\(E\\) is a morphisms of double vector bundles (bi-graded manifolds)<\/p>\n \\(\\Gamma: E \\times_{M}TM \\rightarrow TE\\),<\/p>\n that acts as the identity on the vector bundles \\(E\\) and \\(TM\\).<\/p>\n So, does this reproduce the more standard definitions? Let us look at this in local coordinates;<\/p>\n \\((x^{A}, y^{a}, \\dot{x}^{A}, \\dot{y}^{a})\\circ \\Gamma = (x^{A}, y^{a}, \\dot{x}^{A}, \\dot{x}^{A} y^{b} (\\Gamma_{b}^{\\:\\: a})_{A}(x))\\).<\/p>\n The question now is how do the components of the morphisms \\((\\Gamma_{b}^{\\:\\: a})_{A}\\) transform under changes of local coordinates?<\/p>\n It is not hard to see that we have<\/p>\n \\((\\Gamma_{b’}^{\\:\\: a’})_{A’} = \\frac{\\partial x^{B}}{\\partial x^{A’}}\\left(T_{b’}^{\\:\\: c} (\\Gamma_{c}^{\\:\\: d})_{B}T_{d}^{\\:\\: a’} + T_{b’}^{\\:\\: c} \\frac{\\partial T_{c}^{\\:\\: a’}}{\\partial x^{B}} \\right)\\),<\/p>\n thus the local data of a linear connection as I defined it coincides exactly with the local data of the connection coefficients of connection on a vector as defined in any textbook on differential geometry.<\/p>\n Acknowledgements <\/strong> References<\/strong> [2] J. Grabowski and M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys.<\/em> 62<\/strong> (2012), no. 1, 21-36.<\/p>\n","protected":false},"excerpt":{"rendered":" I will assume the reader has some basic knowledge of differential geometry including the standard notions of a connection. Connections on (or in) fibre bundles are confusing things when you first meet them. They maybe introduced as a way of defining parallel transport of objects along curves, for the tangent bundle you may have been … Continue reading What is a connection on a vector bundle?<\/span>
\nNow let me define in a non-standard way a linear connection.<\/p>\n
\nI won’t claim originality here, this understanding was largely inspired by a lecture given by Prof. Urbanski.<\/p>\n
\n[1] Kirill C. H. Mackenzie, “General Theory of Lie Groupoids and Lie Algebroids”, Cambridge University Press, 2005.<\/p>\n