Notes:An introduction to manifolds - Loring W. Tu

Chapter 1

Section 1: Smooth functions on Euclidean space

1.1: [ilmath]C^\infty[/ilmath] vs Analytic functions

Example:A smooth function that is not real analytic

Section 2: Tangent vectors in [ilmath]\mathbb{R}^n[/ilmath] as Derivations

• [math]D_vf\eq\lim_{t\rightarrow 0}\left(\frac{f(c(t))-f(p)}{t}\right)\eq\frac{d}{dt}f(c(t))\Big\vert_{t\eq 0} [/math]

2.2: Germs of functions

• Equivalence relation yeah yeah yeah
• We define an equivalence relation on the [ilmath]C^\infty[/ilmath] functions defined in some neighbourhood of [ilmath]p\in\mathbb{R}^n[/ilmath].
• Consider the set of all ordered pairs, [ilmath](f,U)[/ilmath], where [ilmath]U[/ilmath] is an open neighbourhood and [ilmath]f:U\rightarrow\mathbb{R} [/ilmath] is a [ilmath]C^\infty[/ilmath] (AKA: Smooth function).
  • We define [ilmath](f,U)\sim(g,V)[/ilmath] if:
    • There exists an open set: [ilmath]W\subseteq U\cap V[/ilmath] with [ilmath]p\in W[/ilmath] such that:
      • [ilmath]f\big\vert_W\eq g\big\vert_W[/ilmath]
• Germ: The equivalence class of [ilmath](f,U)[/ilmath] is called the Germ of [ilmath]f[/ilmath] at [ilmath]p[/ilmath]
• We write the set of all germs of [ilmath]C^\infty[/ilmath] functions on [ilmath]\mathbb{R}^n[/ilmath] at [ilmath]p[/ilmath] as:
  • [ilmath]C^\infty_p(\mathbb{R}^n)[/ilmath] or simply [ilmath]C^\infty_p[/ilmath] if the space is obvious.

Example

Define:

• [math]f:(\mathbb{R}-\{1\})\rightarrow\mathbb{R} [/math] by [ilmath]f:x\mapsto\dfrac{1}{1-x} [/ilmath] and
• [ilmath]g:(0,1)\rightarrow\mathbb{R} [/ilmath] by [math]g:x\mapsto 1+\sum^\infty_{n\eq 1}x^n[/math]

Then [ilmath]f[/ilmath] and [ilmath]g[/ilmath] have the same germ at any point [ilmath]p\in (-1,1)[/ilmath]

Grade: B

There is a topic that needs to be explored here

Some aspect of this work needs exploring and researching in more detail.
The message provided is:

This is an important concept

Then we just hit the concept of an algebra

Section 3: The exterior algebra of multicovectors

3.3: Multilinear functions

• [ilmath]k[/ilmath]-linear function. A multilinear function: [ilmath]f:V^k\rightarrow\mathbb{R} [/ilmath].
• Future: Permutation action: Let [ilmath]f[/ilmath] be [ilmath]k[/ilmath]-linear and let [ilmath]\sigma\in S_k[/ilmath] - the symmetric group on [ilmath]k[/ilmath] symbols. Then:
  • [ilmath](\sigma f)(v_1,\ldots,v_k):\eq f(v_{\sigma(1)},\ldots,v_{\sigma(k)})[/ilmath]
• Symmetric: [ilmath]\forall\sigma\in S_k[\sigma f\eq f][/ilmath]
• Alternating: [ilmath]\forall\sigma\in S_k[\sigma f\eq\text{Sign}(\sigma)f][/ilmath]

Notations

• [ilmath]L_k(V)[/ilmath] - all [ilmath]k[/ilmath]-linear functions
• [ilmath]A_k(V)[/ilmath] - all alternating [ilmath]k[/ilmath]-linear functions.

Lemma 3.11:

• If [ilmath]\sigma,\tau\in S_k[/ilmath] and [ilmath]f[/ilmath] is [ilmath]k[/ilmath]-linear then:
  • [ilmath]\tau(\sigma f)\eq(\tau\sigma)f[/ilmath]

3.5: The symmetrising and alternating operators

Let [ilmath]f\in L_k(V)[/ilmath], then:

• [math]Sf:\eq \sum_{\sigma \in S_k} \sigma f[/math]
• [math]Af:\eq \sum_{\sigma \in S_k} \text{Sign}(\sigma)\sigma f[/math]

Lemma 3.14:

• If [ilmath]f\in L_k(V)[/ilmath] is an alternating [ilmath]k[/ilmath]-linear function already then:
  • [ilmath]Af\eq (k!)f[/ilmath]

3.6: The tensor product

Let [ilmath]f\in L_k(V)[/ilmath] and [ilmath]g\in L_\ell(V)[/ilmath], then their tensor product is a [ilmath](k+\ell)[/ilmath]-linear function, [ilmath]f\otimes g[/ilmath] defined as follows:

• • [ilmath](f\otimes g)(v_1,\ldots,v_{k+\ell}):\eq f(v_1,\ldots,v_k)g(v_{k+1},\ldots,v_{k+\ell})[/ilmath]

3.7: The wedge product

Let [ilmath]f\in A_k(V)[/ilmath] and [ilmath]g\in A_\ell(V)[/ilmath], the wedge product is a product that is alternating also:

• [ilmath]f\wedge g:\eq \frac{1}{k!\ell !}A(f\otimes g)[/ilmath], or explicitly:
• [math]f\wedge g(v_1,\ldots,v_{k+\ell})\eq\frac{1}{k!\ell!}\sum_{\sigma\in S_{k+\ell} } \text{Sign}(\sigma)f(v_{\sigma(1)},\ldots, v_{\sigma(k)})g(v_{\sigma(k+1)},\ldots,v_{\sigma(k+\ell)})[/math]

This is obviously alternating.

Suppose that [ilmath]f(v_1,v_2)g(\text{whatever})[/ilmath] is a term, then so is [ilmath]-f(v_2,v_1)g(\text{whatever})[/ilmath] say too.

Remember [ilmath]f[/ilmath] is alternating by definition, that means:

• [ilmath]f(v_2,v_1)\eq -f(v_1,v_2)[/ilmath]

So we really have [ilmath]2f(v_1,v_2)g(\text{whatever})[/ilmath] in the term. There are a lot of redundancies.

Definition:

• A permutation, [ilmath]\sigma\in S_{k+\ell} [/ilmath] is a [ilmath](k,\ell)[/ilmath]-shuffle if:
  • [ilmath]\sigma(1)<\sigma(2)<\cdots<\sigma(k-1)<\sigma(k)[/ilmath] and [ilmath]\sigma(k+1)<\sigma(k+2)<\cdots<\sigma(k+\ell-1)<\sigma(k+\ell)[/ilmath]

Now we may re-write [ilmath]f\wedge g[/ilmath] as:

• [math](f\wedge g)\eq\sum_{\sigma\ :\ (k,\ell)\text{-shuffle} } \text{Sign}(\sigma)f(v_{\sigma(1)},\ldots,v_{\sigma(k)})g(v_{\sigma(k+1)},\ldots,v_{\sigma(k+\ell)})[/math]

Caveat:All of this is VERY informal... there needs to be proof.... but I'll go along

Section 4: Differential Forms on [ilmath]\mathbb{R}^n[/ilmath]

4.1: Differential [ilmath]1[/ilmath]-forms and the differential of a function

• Cotangent space: to [ilmath]\mathbb{R}^n[/ilmath] at [ilmath]p\in\mathbb{R}^n[/ilmath] is denoted by [ilmath]T^*_p(\mathbb{R}^n)[/ilmath] is defined to be:
  • [ilmath](T_p(\mathbb{R}^n))^*[/ilmath] - or [ilmath](T_p(\mathbb{R}^n))^\vee[/ilmath] in this author's notation - of the tangent space [ilmath]T_p(\mathbb{R}^n)[/ilmath]