What is a connection on a vector bundle?

April 5th, 2014 by ajb

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 introduced to them as directional derivatives and so on. A rather general notion is that put forward by Ehresmann as the compliment to the vertical bundle which we call the horizontal bundle. Note that Ehresmann’s defintion works for any fibre bundle and not just vector bundles. Even more abstractly we can define a connection on a fibred manifold as a section of its first jet bundle.

Okay, so we all agree that are confusing things. However, for vector bundles we can describe them completely geometrically as certain maps between double vector bundle. For double vector bundles see for example [1]

Double vector bundles
I won’t describe general double vector bundles here, all we really need is the tangent bundle of a vector bundle $latex TE $, which we equip with natural local coordinates $latex (x^{A}, y^{a}, \dot{x}^{A}, \dot{y}^{a}) $. The coordinate transformations here are given by

$latex x^{A’} = x^{A’}(x)$
$latex y^{a’} = y^{b}T_{b}^{\:\: a’}(x)$
$latex \dot{x}^{A’} = \dot{x}^{B}\frac{\partial x^{A’}}{\partial x^{B}}(x)$
$latex \dot{y}^{a’} = \dot{y}^{b}T_{b}^{\:\: a’}(x) + y^{b}\dot{x}^{B}\frac{\partial T_{b}^{\:\:a’}}{\partial x^{B}}(x)$

It is now convenient to view this double vector bundle as a bi-graded manifold by assigning a bi-weight as $latex w(x) = (0,0)$, $latex w(y) = (1,0)$, $latex w(\dot{x}) = (0,1)$ and $latex w(\dot{y}) = (1,1)$.

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.

We have the following vector bundle structures


The core of this double vector bundle is identified with $latex ker(T\pi)$ which is the vertical bundle $latex VE$, which you can easily see is isomorphic to $latex E$.

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

$latex TE \rightarrow E \times_{M} TM \rightarrow M $,

defined by first projecting out $latex \dot{y}$, which has weight 2 and the then projecting out $latex y$ and $latex \dot{x}$ which have weight 1. Again you can see that this is all consistent with the coordinate transformations.

Also note that $latex E \times TM$ also carries the structure of a double vector bundle simply by projection onto the two factors and then the obvious projections to $latex M$. This can be viewed keeping the bi-weight as inherited by that on $latex TE $.

Connection in a vector bundle
Now let me define in a non-standard way a linear connection.

Def A (linear) connection on a vector bundle $latex E$ is a morphisms of double vector bundles (bi-graded manifolds)

$latex \Gamma: E \times_{M}TM \rightarrow TE$,

that acts as the identity on the vector bundles $latex E$ and $latex TM$.

So, does this reproduce the more standard definitions? Let us look at this in local coordinates;

$latex (x^{A}, y^{a}, \dot{x}^{A}, \dot{y}^{a})\circ \Gamma = (x^{A}, y^{a}, \dot{x}^{A}, \dot{x}^{A} y^{b} (\Gamma_{b}^{\:\: a})_{A}(x))$.

The question now is how do the components of the morphisms $latex (\Gamma_{b}^{\:\: a})_{A}$ transform under changes of local coordinates?

It is not hard to see that we have

$latex (\Gamma_{b’}^{\:\: a’})_{A’} = \frac{\partial x^{B}}{\partial x^{A’}}\left(T_{b’}^{\:\: c} (\Gamma_{c}^{\:\: d})_{B}T_{d}^{\:\: a’} + T_{b’}^{\:\: c} \frac{\partial T_{c}^{\:\: a’}}{\partial x^{B}} \right)$,

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.

I won’t claim originality here, this understanding was largely inspired by a lecture given by Prof. Urbanski.

[1] Kirill C. H. Mackenzie, “General Theory of Lie Groupoids and Lie Algebroids”, Cambridge University Press, 2005.

[2] J. Grabowski and M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), no. 1, 21-36.

The 2014 Abel Prize

April 4th, 2014 by ajb
The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2014 to Yakov G. Sinai of Princeton University and Landau Institute for Theoretical Physics, Russian Academy of Sciences for “his fundamental contributions to dynamical systems, ergodic theory, and mathematical physics”.

Sinai is best known for his works finding strong links between dynamical systems (deterministic) and stochastic systems (probabilistic).

The Abel Prize website.

Warning about UK science spending

April 1st, 2014 by ajb
The science board of the Science and Technology Funding Council (STFC) published a report last week warning about the effects of the flat cash funding of UK science.

According to the report the UK spent 1.8% of GDP on research and development over the past twenty years. The average spend of comparator countries has been 2.9%.

Other countries have increased their spending. For example South Korea has doubled its spending over the last few years to 4% GDP

The bottom line is unless we spend more money on science there could be huge damage to the scientific standing of the UK in the future, which at the moment punches well above its weight.

Stark warnings on spending made by STFC’s Science Board IOP website.

The Big Bang Fair South Wales

March 17th, 2014 by ajb
flag of wales The Big Bang Fair South Wales will be held at the Celtic Manor resort Newport on the 7th of April 2014. The fair is an interactive engineering and science event designed to show young people that STEM subjects can be exciting.

The event is aimed at 9 to 18 year olds and opens at 10.30am with a 3.30pm closing time.

To find out more details follow the link below.

The Big Bang Fair South Wales

Charlotte Church to study physics

March 13th, 2014 by ajb

Image by BrotherDarksoul
The Welsh singer Charlotte Church has said she wants to obtain a degree in physics.

Speaking to BBC Wales she said

I want to go and do a degree in physics – I will have to do an A-level in physics and maths first though.

What ever she decides I wish her luck in her studies.

The other music stars turned scientists include Brian May of Queen fame and Brian Cox.

Charlotte Church plans to study for degree in physics BBC News

Do we need more STEM graduates?

March 12th, 2014 by ajb
The Confederation of British Industry (CBI) suggest that the government should make careers in science, technology, engineering and maths (STEM) more attractive and that one method could be to reduce the tuition fees.

More STEM graduates?

A CBI/Pearson survey suggests that 42% of UK firms faced difficulties recruiting individuals with STEM skills and knowledge last year.

Highly-skilled workers are essential for our growth sectors and it will be those young people with science and maths who will go on to become the engineers and new tech entrepreneurs of tomorrow.

Katja Hall, the CBI’s chief policy director

The other side of the coin
Call me skeptic, but my personal observations, as well as talking to others suggests this is simply BS. We don’t in fact have enough STEM jobs in the UK as it is!

Research conducted last year by the employment consultancy Work Communications showed the number of places on all graduate schemes offered by UK employers in all sectors to be 65,000 for the academic year 2012-2013. However, 132,790 UK students graduated with a first degree in STEM subjects in 2011-2012, according to the Higher Education Statistics Agency.

As soon as you look at the numbers, it is very hard to justify [claims of] a skills shortage

Marcus Body, head of research at Work Communications

Again this is backed up with my own anecdotal observations.

Unemployment of STEM graduates

Katherine Sellgren (BBC), in 2011 wrote about Engineering graduates specifically, see here.

Nearly a quarter of UK engineering graduates are working in non-graduate jobs or unskilled work such as waiting and shop work, a report suggests.

Katherine Sellgren (BBC)

Has the situation changed that drastically in the past couple of years? I doubt it.

What is going on?
I have no idea. It seems that we have lots of rather contradictory “evidence” here. One lobbing group says we need more STEM graduates while the unemployment levels and the level of “unsuitable” jobs taken up seems to suggest quite the opposite.

The higher up the education tree the worse it gets..

Does the UK really need more engineers? Times Education website

CBI call to cut tuition fees to end ‘skills vacuum’

Chinese teachers to run “maths hubs” in English schools

March 12th, 2014 by ajb
S Up to 60 mathematics teachers from Shanghai will be brought in to England to help raise mathematics standards in an exchange program to be organised by the Department of Education.

We have some brilliant maths teachers in this country but what I saw in Shanghai – and other Chinese cities – has only strengthened my belief that we can learn from them.

Elizabeth Truss, Education Minister.

This follows her recent visit to China in search of why they do so well in mathematics. I posted about this here.

This is also some worry about mathematics education and attitudes in Wales. I posted about this recently here. With that, I have no idea is the Welsh Assembly will seek to do something similar with Welsh schools.

Is the problem with mathematics education that bad?

I will answer that with a quote…

78% of working-age adults have maths skills below the equivalent of a GCSE grade C – and that half only have the maths skills of a child leaving primary school.

Mike Ellicock, chief executive of National Numeracy

Oh dear…

Shanghai teachers flown in for maths BBC News website

National Numeracy

Geometry seminar at Gdańsk

March 5th, 2014 by ajb
Gdansk I will be giving a talk at the Geometry Seminar of the Institute of Mathematics at Gdansk University on the 19th March.

Title: On the higher order tangent bundles of a supermanifold.

Abstract: In this talk I will introduce the notion of a supermanifold as a “manifold” with both commuting and anticommuting coordinates. After that I will discuss the notion of a curve on a supermanifold and how to construct their jets, this will lead to a kinematic definition of the higher order tangent bundles of a supermanifold. However, to formulate this properly we are lead to more abstract ideas that have their roots in algebraic geometry and in particular notion of internal Homs objects. I will not assume the audience to be experts in the theory of supermanifolds nor jets, only that they have some basic knowledge of standard differential geometry.

Geometry Seminar Gdansk

Welsh parents are urged to be positive about maths

March 3rd, 2014 by ajb
flag of wales A campaign is going to be launched urging parents to have a more positive attitude towards mathematics and learning mathematics. This comes after a poll found three in 10 parents in Wales questioned admitted being negative about maths in front of their children.

Maths results are the worst of any core subject in Wales behind English, Welsh and science.

It’s fair to say that maths is suffering from an image problem, and as today’s poll demonstrates, there is still work to do to change the view in some quarters which is that maths isn’t really important and that it doesn’t really matter what we say to children about it.

We understand the value of strong numeracy skills, for life and for employment.

Education Minister Huw Lewis

The “What you say counts” campaign will be launched by the minister at a supermarket in Cardiff Bay on Monday.

Parents ‘must be positive about maths’, says Huw Lewis BBC News.

Improper use of school computers

March 2nd, 2014 by ajb

I am shocked. After the course, the teacher immediately came to me and told me he had watched porn on a school computer.

René Portenier, director KV Zurich Business School.

But that is not the end of it…he forgot to unplug the cable to the projector!

Professor accidentally beams X-rated porn film on class projector