My Differential Geometry Images

So I am writing some notes on differential geometry, and using metapost for tricky diagrams. Here are a few of the tricky diagrams I have.

The first three images are:

The third image is a tad bit bigger:

Note the fonts in this picture are messed up, since I had to use postscript fonts. It looks far more beautiful in TeX using metapost, trust me!

numeric u;
color yellow;
u := 1pc;
yellow = red+green;

verbatimtex \input amssym.tex etex;

% sphere
beginfig(0)
  draw fullcircle scaled 4u;
  draw (-2u,0)..(0,-.7u)..(2u,0);
  draw (-2u,0)..(0,.7u)..(2u,0) dashed evenly;
endfig;

% plane + cylinder
beginfig(1)
  draw (0,-u)--(0,3u)--(u,4u)--(u,0)--cycle;

  draw (3u,0)..(4u,-.5u)..(5u,0);
  draw (3u,0)..(4u,.5u)..(5u,0) dashed evenly;

  draw (3u,4u)..(4u,4.5u)..(5u,4u);
  draw (5u,4u)..(4u,3.5u)..(3u,4u);

  draw (3u,0)--(3u,4u);
  draw (5u,0)--(5u,4u);
endfig;

% saddle
beginfig(2)
  path p[];
  z[0] = (2u,4u);
  z[1] = (2.5u,3u);
  z[2] = (1.5u,2.5u);
  z[3] = (4u,0u);
  z[4] = (6.5u,u);
  z[5] = (7.u,1.75u);
  z[6] = (6.75u,0.5u);

  p[0] = z[0]..z[1]..z[2];
  p[1] = z[1]..z[3]..z[4];
  p[2] = z[5]..z[4]..z[6];

  p[3] = z[0]--(point 0.75*length(p[1]) of p[0]);
  p[4] = z[2]--(1.75u,-2u);
  p[5] = z[6]--(6.5u,-2u);
  p[6] = z[5]--(point 0.9*length(p[1]) of p[2]);
  for i=0 upto 6:
    draw p[i];
  endfor;

  draw (4u,-u){up}..(point 0.6*length(p1) of p1);
  draw (point 0.6*length(p1) of p1)..{down}(5.125u,-.75u) dashed evenly;
  draw (1.75u,-2u)..(4u,-u)..(6.5u,-2u);

  p7 = ((0,0)--(2u*unitvector(direction 0 of p[6]))) shifted point (length p6) of p[6];
  draw p[7] dashed evenly;

  p8 = ((0,0)--(3.5u*unitvector(direction 0 of p3))) shifted point (length p3) of p3;
  draw p8 dashed evenly;

  p9 = (point (length p8) of p8)..(5.125u,-.75u)..(point (length p7) of p7);
  draw p9 dashed evenly;

endfig;

beginfig(3)
  picture Rn;
  picture sphere;
  picture torus;
  % R^n
  Rn = image(
      for i=1 upto 3:
      draw (-.5u,i*u)--(3.5u,i*u) withcolor 0.75white;
      draw (i*u,-.5u)--(i*u,3.5u) withcolor 0.75white;
    endfor;
    drawdblarrow (-.5u,0)--(3.5u,0);
    drawdblarrow (0,-.5u)--(0,3.5u);
    );
  % sphere  
  sphere = image( 
      draw fullcircle scaled 4u;
    draw ((-2u,0)..(0,-.7u)..(2u,0));
    draw ((-2u,0)..(0,.7u)..(2u,0)) dashed evenly;
    );
  % torus
  torus = image(
      path hole;
    path uhole;
      draw
      (0,2u)..(-2u,0)..(0,-2u)..(2u,-u)..(4u,-2u)..(6u,0)..(4u,2u)..(2u,u)..cycle;
    hole = halfcircle rotated 180 scaled 2u shifted (4.5u,0);
    draw hole;
    uhole = (point 0.1*length(hole) of hole)
     ..(0.5[(point 0.1*length(hole) of hole),(point 0.9*length(hole) of hole)]+(0,.5u))
     ..(point 0.9*length(hole) of hole);
    draw uhole; 
    );
  
  draw Rn;
  draw Rn shifted (20u,-5u);

  draw torus shifted (8u,4u);

  path localPatch;
  path imageOfF;
  path preimageOfG;
  path imageOfG;
  path imageOfPsi;
  path intersection;
  localPatch = (fullcircle scaled 2u shifted (1.5u,1.5u));
  fill localPatch withcolor 0.75[blue,white];
  draw localPatch dashed evenly;

  z[0] = (((u,0)--(u,2u)) intersectionpoint localPatch);
  draw (((0,u)--(u,u)) intersectionpoint localPatch)
  --(((2u,u)--(4u,u)) intersectionpoint localPatch) withcolor 0.75[black,blue];
  draw (((0,2u)--(2u,2u)) intersectionpoint localPatch)
  --(((2u,2u)--(4u,2u)) intersectionpoint localPatch) withcolor 0.75[black,blue];
  draw (((u,0)--(u,2u)) intersectionpoint localPatch)
  --(((u,2u)--(u,4u)) intersectionpoint localPatch) withcolor 0.75[black,blue];
  draw (((2u,0)--(2u,2u)) intersectionpoint localPatch)
  --(((2u,2u)--(2u,4u)) intersectionpoint localPatch) withcolor 0.75[black,blue];

  imageOfF = fullcircle xscaled 3u yscaled 1.5u shifted (8u,4u);
  fill imageOfF withcolor 0.75[blue,white];
  draw imageOfF dashed evenly;

  picture preG;

  preG = image(
  
  preimageOfG = unitsquare scaled 2u rotated 45 shifted (22u,-5u);

    fill preimageOfG withcolor 0.5[yellow,white];
    fill fullcircle xscaled u yscaled 5u rotated -23 shifted (22u,-3u)
      withcolor 0.5[green,white];
      draw (((21u,-4u)--(21u,-5u)) intersectionpoint preimageOfG)
    --(((21u,-5u)--(21u,-3u)) intersectionpoint preimageOfG)
    withcolor 0.5[black,yellow];
  draw (((22u,-2u)--(22u,-3u)) intersectionpoint preimageOfG)
    --(((22u,-4u)--(22u,-8u)) intersectionpoint preimageOfG)
    withcolor 0.5[black,yellow];
  draw (((23u,-3u)--(23u,-3.5u)) intersectionpoint preimageOfG)
    --(((23u,-3.5u)--(23u,-8u)) intersectionpoint preimageOfG)
    withcolor 0.5[black,yellow];
  draw (((21u,-4u)--(22u,-4u)) intersectionpoint preimageOfG)
    --(((22u,-4u)--(25u,-4u)) intersectionpoint preimageOfG)
    withcolor 0.5[black,yellow];
  draw (((21u,-3u)--(22u,-3u)) intersectionpoint preimageOfG)
    --(((22u,-3u)--(25u,-3u)) intersectionpoint preimageOfG)
    withcolor 0.5[black,yellow];
  draw preimageOfG dashed evenly;

  clip currentpicture to preimageOfG;
    );
  draw preG;

  imageOfG = (19u,2.5u)--(20u,3u)--(21.5u,3u)--(21u,2.5u)--cycle;


  fill imageOfG withcolor 0.5[white,yellow];

  imageOfPsi = fullcircle xscaled 2u yscaled u rotated -45
    shifted (20u,3u);
  fill imageOfPsi withcolor 0.75[blue,white];


  numeric t[];
  (t[0],whatever) = imageOfPsi intersectiontimes ((19u,2.5u)--(20u,3u));
  (t[1],whatever) = imageOfPsi intersectiontimes ((21u,2.5u)--(19u,2.5u));
  (t[2],whatever) = imageOfPsi intersectiontimes ((20u,3u)--(23u,3u));  
  (t[3],whatever) = imageOfPsi intersectiontimes ((19u,2.5u)--(20u,2.5u));
  
  intersection = (subpath(t0,t3) of imageOfPsi)--%(20u,3u)--
    (subpath(t1,t2) of imageOfPsi)--(20u,3u)--cycle;
  
  fill intersection withcolor 0.5[green,white];
  draw imageOfPsi dashed evenly;
  draw imageOfG dashed evenly;
  draw sphere shifted (20u,4u);

  p[0] := (point 0.25*length(localPatch) of localPatch)..(4u,4u)
  ..(point 0.5*length(imageOfF) of imageOfF);
  p[1] := (point 0 of imageOfF)..(12u,5u)..(19.5u,3.5u);
  p[2] := (21.5u,3u)..(23u,0)..(23u,-3u);
  p[3] := (point .75*length(localPatch) of localPatch)..(1.5u,-u)
  ..(10u,-4u)..(21.5u,-4.5u);
  drawarrow p[0];
  
  drawarrow p1;

  drawarrow p2;

  drawarrow p3 dashed evenly;

  label.top(btex $x$ etex, point 0.5*length(p0) of p0);
  label.top(btex $\varphi$ etex, point 0.75*length(p1) of p1);
  label.rt(btex $y^{-1}$ etex, point 0.5*length(p2) of p2);
  label.bot(btex $y^{-1}\circ\varphi\circ x$ etex, point 0.6*length(p3) of p3);

  label(btex $U$ etex, (1.5u,1.5u)) withcolor 0.5[black,blue];

  draw (8u,4u) withpen pencircle scaled 3;
  label.lft(btex $p$ etex, (8u,4u)) withcolor 0.75[blue,black];
  label.bot(btex $M$ etex, (12u,2u));

  label.rt(btex $N$ etex, (22u,4u));

  label.lft(btex $\Bbb{R}^{n}$ etex, (20u,-2u));

  label.lft(btex $\Bbb{R}^{m}$ etex, (0,3u));
endfig;


end;

Outline of Commutative Geometry

So, preparing for my discussion of noncommutative geometry, I need to discuss "commutative geometry".

What's going on here? Well, lets begin with the simplest notion of a space: a topological space.

I have discussed in my notebk [googlecode.com] the notion of a topological space and continuous functions.

However, the algebra of continuous real-valued (or, more generally, complex-valued) functions encode the topology.

So I need to write notes reconstructing the topological data for $X$ from the ring structure and properties which $C(X)$ satisfy.

Vector Bundles

The next sort of space we can work with is a vector bundle. What's this guy?

Well, it's really a fibre bundle whose fibre forms a vector space. What's a fibre bundle?

It's a generalization of the product space where we fix one of the spaces.

Where does this occur? In vector calculus!

We are working with $\mathbb{R}^{3}$. A vector field assigns to each point in $\mathbb{R}^{3}$ a vector. But vectors live in "linear spaces" (or vector spaces).

So secretly we have $X=\mathbb{R}^{3}$ be the underlying space, and the total space be $E=F\times X$ consisting of "tangent vectors" (an ordered pair consisting of the vector assigned by the vector field, and its base point).

The fibre here is a vector space. Moreover, it is $F=\mathbb{R}^{3}$ as a vector space.

This is the simplest example of a vector bundle. So what?

Well, vector fields can be represented through ordered triples. That is, three smooth functions represent each component of the vector field (the x part, y part, and z part).

So algebraically we have $C^{\infty}(\mathbb{R}^{3})\times C^{\infty}(\mathbb{R}^{3})\times C^{\infty}(\mathbb{R}^{3})$ represent all possible vector fields on our space.

This is a free module over $C^{\infty}(\mathbb{R}^{3})$. So are vector bundles represented by free modules?

Not really, we use projective modules (which is more general).

There are a few other things to discuss on this matter, e.g., global sections, and so forth.

Spinor Bundles

This should be discussed in some detail, as there are few good references on the subject.

Even nLab's entry on Spinor bundles is lacking, alas!

However, we need to encode this data in a spectral triple. See the nLab's entry, it is quite good.

See also Alain Connes' "On the spectral characterization of manifolds" (arXiv:0810.2088) for details.

This would take some time to write up.

Noncommutative Rejoinder

I suspect that by taking an arbitrary ring, instead of the ring of smooth functions (or continuous functions, or...), we begin working with noncommutative geometry.

Is this algebraic geometry? No, not really. Algebraic geometers use polynomials to encode their geometric objects.

On the other hand, what we are doing here is considering the structure of rings and modules over our rings to encode geometric properties and data.

Of course, I may be misinformed on what algebraic geometers do...I frankly never understood it well enough to satisfy myself.

Ancient Geography

Reading Herodotus requires a greater knowledge of geography than I previously implied.

Fortunately, all the geography you need to know is encoded in a free map. Ironically it comes from a video game titled "Rome: Total Realism".

So, without further ado, here is the map:

Does it seem small? Well, it's a link, so click on it for the huge map.

Experiments as "registers" in notes

Mathematics is about theorems, science is about experiments. But theorems are expressed rigorously in formal logic...what about experiments?

The "old fashioned" way would be to write up a lab report and carry out the experiment! However, nowadays, there are some programs to simulate experiments.

For example, the KDE Education Project has a few programs for physics, astronomy, and chemistry. Sadly, no biology (although I wouldn't know what that would look like...).

Additionally, if one is writing notes on history, there is a beautiful globe program one could use...

At any rate, for classical physics, this is perhaps the ideal set up. It allows the reader to play with the experiment, which is the entire idea of an experiment(!), and if so possessed one could set it up in real life.

Plus this is free, a small perk.

Physics

There is the Step program which is useful for classical mechanics.

Better, the physical systems being studied can be represented as a program.

And when you get to numerical analysis, you can perform a "How accurate was this?" series of exercises ;)

It is multiplatform, so anyone on Windows, Mac OS X, or *nix can use the KDE Education suite. It may be worth while to consider :)

However, I wonder if there is not a better program out there? The qt framework is always a wee bit heavy duty.

It would be nice if someone invented a scripting language that produced flash videos, or something along those lines...since qt runs slow on my 733MHz Pentium 3 processor :'(

Chemistry

I have not thought about writing up chemistry notes, although I feel if I write biology notes then it would become necessary to study chemistry.

Personally, chemistry seems to be a sibling to physics, it just is concerned with different scales. So experiment is very important in chemistry, and it is just as repeatable as in physics.

The Kalzium suite simulates chemistry to a degree. Everything is always "to a degree"!

Biology?

I am unaware of any similar such program for biology...but there are few experiments which are reproducible in biology!

Perhaps something like "virtual fruit flies" or "virtual peas" would be nice to study genetics, but I think it would be best to do it the "old fashioned way" (i.e., actually carry out the experiment!).

Take Home Message

If you are a mathematician writing notes on science, and are unable to perform the experiments: write a program.

Math to think about

There are several interesting directions I'd like to investigate. So interesting, I have decided to let you in on it too!

Moonshine

No living man cannot deny interest in moonshine. Terry Gannon's "Monstrous moonshine and the classification of CFT" (arXiv:math/9906167) provides a great review.

The basic idea is that we have a way to associate to "algebraic stuff" (e.g., groups) some "modular stuff".

What's great is, the word "stuff" is used in the technical sense of the word.

This would require reviewing group theory, finite groups, as well as some ring theory (for the "algebraic stuff"). To discuss "modular stuff", we'd need to review complex analysis, functional analysis. The connection here would require studying conformal field theory, to some degree.

Robert Wilson's Finite Simple Groups is a wonderful reference for finite simple groups; and as always SPLAG is a good reference too.

Noncommutative Geometry

People mean many things by "Noncommutative Geometry", here I mean Connes' approach.

I suppose this first requires us to consider what "commutative geometry" is!

This requires knowledge of commutative algebra and differential Geometry. The idea is to model "differential calculus over a commutative ring" (as Wikipedia calls it), i.e., consider the algebraic "grammar" underlying differential geometry.

Commutative algebra describes this algebraic "grammar". Studying this model in commutative algebra is precisely what I mean by "commutative geometry" (where projective modules correspond to vector bundles, and so on).

Noncommutative geometry, on the other hand, generalizes this model to the noncommutative setting!

I still need to write up my notes on differential geometry, but there are no good references for "commutative geometry"! There are a few books on commutative algebra, though...

I should type up my notes on algebraic topology too, since spin bundles are a "principal Spin bundle". Although I have some notes written on the Spin group (see my Lie groups notes), I should review it some more. Michelson and Lawson's Spin Geometry is a wonderful book to consider...

Operator algebras need to be reviewed for considering spectral triples. The algebra we typically work with are von Neumann algebras which are related to C* algebras.

Some references for operator algebras:

1. Kadison and Ringrose, Fundamentals of the theory of operator algebras vol. I and II
2. Blackadar. Operator Algebras: Theory of C*-Algebras and von Neumann Algebras. Encyclopaedia of Mathematical Sciences. Springer-Verlag, 2005.
3. Yasuyuki Kawahigashi, "Conformal Field Theory and Operator Algebras" arXiv:0704.0097 (18 pages)
4. Meghna Mittal, Vern Paulsen, "Operator Algebras of Functions." arXiv:0907.5184
5. John M. Erdman Lecture Notes on Operator Algebras (129 pages)
6. J. A. Erdos, C*-Algebras (51 pages).
7. N.P. Landsman, "Lecture notes on C*-algebras, Hilbert C*-modules, and quantum mechanics" arXiv:math-ph/9807030 (89 pages).
8. Jacob Lurie's Course Notes on Von Neumann Algebras, quite comprehensive!
9. Wassermann, Operators on Hilbert space [ps] (70 pages)
10. VFR Jones, von Neumann algebras [pdf] (150 pages)
11. NP Landsman's Lecture Notes on Operator Algebras [pdf] (64 pages).
12. John Hunter and Bruno Nachtergaele, Applied Analysis (free, legal ebook!)

And, of course, there is Connes' Noncommutative Geometry [pdf], as well as Connes and Marcolli's Noncommutative Geometry, Quantum Fields and Motives [pdf]