More about connections: a toy example

In this post we are going to work out the concepts of the previous entry to develop the intuition of the ideas.

Let's take \(M=\mathbb{R}\) and \(E=M \times \mathbb{R}=\mathbb{R}\times \mathbb{R}\). Smooth sections of this bundle \(\pi: E\mapsto M\) can be identified with smooth functions
\[
f:\mathbb{R}\longmapsto \mathbb{R}
\]
if we take the frame given by the constant section \(e(x)=(x,1)\). That is, a section would be
\[
\sigma(x)=f(x) e(x)
\]
where \(f(x)\) is any smooth function.


Suppose we have a generic connection defined on \(E\), \(\nabla\). We have
\[
\nabla(\sigma)=e\otimes df+e\otimes f(x) \omega(x)
\]
being \(\omega\) a 1-form, the connection form. With this set up, the usual derivative correspond to a connection whose connection form is \(\omega=0\). In effect, for the vector field \(\partial x\) on \(M\) we have
\[
\nabla(\sigma)(\partial x)=df(\partial x) e+0=f'(x) e
\]



But now, let's take a different \(\omega\) to obtain a new notion of derivative. For example, we can choose \(\omega(x)=2dx\). In this case, when we derive the section \(\sigma\) we obtain
\[
\nabla(\sigma)(\partial x)=df(\partial x) e+f(x)2dx(\partial x)e=(f'(x)+2f(x))e
\]

So this new notion of derivative associates
\[
f(x)\longmapsto f'(x)+2f(x)
\]

What functions would correspond to constant functions? Well, those whose derivative were 0, that is
\[
f'(x)+2f(x)=0
\]

Solving this ODE we obtain the family
\[
f(x)=Ke^{-2x}
\]
that corresponds to constant functions.


Let's stop. What's going on here? When we learn to derive usual functions
\[
f:\mathbb{R}\longrightarrow \mathbb{R}
\]
we assume that the arriving \(\mathbb{R}\) is always the same. But in a more general sense, we can think that every \(f(x)\) lives in a different space \(E_x=\mathbb{R}\), or the same space but measured with different units. For example, imagine that we live in a 1 dimensional world, and want to study the movement of a particle that is initially at 1 meter from us. To do this, we check the distance every 1 second and annotate it in a table. We say that the particle is not moving if its distance is always 1.


But now, imagine that the stick we use to measure the distance is increasing its size (maybe because of rising temperatures, who knows). For example, suppose that the size of the stick is \(e^{2t}\), where \(t\) is time. Then a distance \(d\) measured at time 0 would correspond to a distance \(d e^{-2t}\) measured at time \(t\) (think about it). Therefore, we would consider that the particle is at rest (constant position function) when its curve is of the form
\[
f(t)=de^{-2t}
\]





These lines are interpreted as joining point in \(\mathbb{R}\) that can be identified as the same. They act like parallel transporters. Their tangent lines define the horizontal subbundle of \(TE\).

Comments

Post a Comment

Popular posts from this blog

About the definition of angle, from Geometric Algebra

Image
In GA, bivectors represent  2-dimensional directions. Any simple bivector can be written as \[ \theta e_1e_2 \] with \(\theta \in \mathbb{R}\) and \(\{e_i\}\) unitary and orthogonal vectors. The unitary vectors specify the 2- direction itself, and \(\theta\) is the size (analogous to the length of a vector, that is a 1-dimensional direction). Given two vectors \(a\) and \(b\) (that we take to be unitary, wlog) we say that the simple bivector \(\theta e_1e_2\) is the angle formed by them if \[ ab=e^{\theta e_1 e_2} \] where we take as a definition \[ e^A=1+A+A^2 /2+\cdots=\lim_N (1+\frac{A}{N})^N \] Real analysis let us assure that this expression is well defined when \(A\) is a 2-blade. In fact, since \(e_1e_2e_1e_2=-1\), we can write \[ e^{\theta e_1 e_2}=cos(\theta)+e_1 e_2sin(\theta) \] where \(cos\) and \(sin\) are defined by their power series. Oserve that \(b=a e^{\theta e_1 e_2}\) and then \[ b=a(cos(\theta)+e_1 e_2sin(\theta))=a\cdot cos(\theta)+ ae_1e_2 \cdot sin(\theta) \]
Read more

Spherical, hyperbolic and parabolic geometry

Image
Let's consider the space \(\mathbb{M}^4\), that is, the manifold \(\mathbb{R}^4\) with coordinates \((x,y,z,t)\) but with the pseudo-Riemannian metric given in each tangent space by \[ ds^2=dx^2+dy^2+dz^2-dt^2. \] This metric is left invariant by the group \(O(3,1)\) when acting on \(\mathbb{M}^4\). It also leaves invariant the cone \[ t^2-x^2-y^2-z^2=0 \] so is logical that we pay attention to it. Now, following TRTR from Penrose page 423, we can consider the family of hyperplanes inside \(\mathbb{M}^4\) given by \[ z+t+\lambda(t-z)=2 \] intersecting our cone in different 2-dimensional submanifolds \(P_{\lambda}\) of \(\mathbb{M}^4\). Here we have a picture from that book, where we consider only \((x,z,t)\) for purposes of drawability. If we analyse the case \(\lambda=0\), we observe that we obtain the submanifold \(P_0\), \(E\) in the picture with a shape of a parabola, given by the embedding \begin{align*}   \phi \colon \mathbb{R}^2 &\longrightarrow \mathbb{M}^4\\   (u_1,u_2
Read more

4-dimensional spheres, or haunting an ant

Image
In this post, we will try to explain a little bit what it feels like to live in a 4-dimensional world. First, we will see that we can enter the inside of a normal sphere without breaking the wall. And secondly, we will try to describe what we would feel if our world were not the classic 3D universe, known as \(\mathbb{R}^3\), but another 3-dimensional space called \(S^3\). But before we start, let's name things. The circles we all know have a more technical name for mathematicians: \(S^1\). On the other hand, a sphere (let's say, a ball) is called \(S^2\). This similarity in names is not accidental: since we were children, we intuitively knew that circles and spheres have something in common. In fact, they are "the same" except for the detail that circles are 1-dimensional (you can only move forward or backward), and spheres are 2-dimensional (if you lived in one, you could move forward-backward, but also left-right. Oops, actually, we live in one). Now let's
Read more