Differential Geometry Series Part 2

Euclidean Geometry

In the previous chapter, we touched on the history of Differential Geometry and discussed what we would be doing in this series. In this chapter, we will start from the very foundations of differential geometry and introduce some of the mathematical structures we will need in order to do Differential Geometry. If this runs too long, we will continue introducing these structures in the next chapter. First and foremost, to do differential geometry, we need the structure of a vector space. So cue the music, Uğur: Vector Spaces.

1. Vector Spaces

I am now going to cheat a little and assume that you know at least a bit of linear algebra. When you hear the term vector space, the vectors you encountered in physics class may come to mind; what we will discuss here, however, is a generalized version of that familiar notion of a vector. With suitably defined operations, for example, even a polynomial can be regarded as a vector.

Definition 1. Let $V$ be a set and let $(+, \ast)$ be binary operations. If the following conditions are satisfied, we call the triple $(V, +, .)$ a vector space.

1. (Closure under addition) For all $u, v \in V$, $u + v \in V$
2. (Commutativity under addition) For all $u, v \in V$, $u + v = v + u$
3. (Associativity) For all $u, v, w \in V$, $(u + v) + w = u + (v + w)$
4. (Existence of the zero vector) There is a distinguished vector $0_V \in V$ satisfying $u + 0_V = u$ for every $u \in V$.
5. (Existence of an additive inverse) For every $u \in V$, there is a $v \in V$ satisfying $u + v = 0_V$. In this case, we write $v = -u$.
6. (Closure under multiplication) For all $c \in \mathbb{R}$ and all $u \in V$, $c \ast u \in V$.
7. (Distributivity over scalar addition) For all $c, d \in \mathbb{R}$ and all $u \in V$, $(c + d) \ast u = c \ast u + d \ast u$.
8. (Distributivity over vector addition) For all $c \in \mathbb{R}$ and all $u, v \in V$, $c \ast (u + v) = c \ast u + c \ast v$.
9. (Associativity) For all $c, d \in \mathbb{R}$ and all $u \in V$, $(c \ast d) \ast u = c \ast (d \ast u)$.

To summarize the definition above, take a set and define on it two operations called “addition” and “multiplication.” If the properties above are satisfied, the set and operations we have placed in the mixing bowl give us a “cake”—that is, a vector space. A few examples should help the ideas settle into place.

Example 1. Let us begin with one of the best-known examples, a space familiar from physics classes.

$$\mathbb{R}^2 = \left\{ \begin{bmatrix}u_1\\u_2\end{bmatrix}\ \Big|\ u_1,u_2\in\mathbb{R} \right\}$$

Let us combine this set with the standard addition and multiplication operations defined below:

$$\begin{bmatrix}u_1\\u_2\end{bmatrix} + \begin{bmatrix}v_1\\v_2\end{bmatrix} = \begin{bmatrix}u_1+v_1\\u_2+v_2\end{bmatrix}, \quad \forall \begin{bmatrix}u_1\\u_2\end{bmatrix}, \begin{bmatrix}v_1\\v_2\end{bmatrix} \in \mathbb{R}^2$$ $$c\begin{bmatrix}u_1\\u_2\end{bmatrix} = \begin{bmatrix}cu_1\\cu_2\end{bmatrix}, \quad \forall c\in\mathbb{R}, \forall \begin{bmatrix}u_1\\u_2\end{bmatrix} \in \mathbb{R}^2$$

Together with these operations, the set $\mathbb{R}^2$ forms a vector space. We can generalize this set. Suppose we define a set $\mathbb{R}^n$ whose elements are vectors with $n$ components, define addition by adding corresponding components, and define multiplication componentwise (I am putting this into words because writing it all out would be rather confusing). This space is also a vector space.

Example 2. (For readers who have taken calculus and analysis courses)

$$C[a,b]=\{f:[a,b]\to\mathbb{R}\mid f \text{ sürekli bir fonksiyon.}\}$$

On this set, define the operations

• $(f+g)(x)=f(x)+g(x),\ \forall x\in[a,b]$
• $(cf)(x)=cf(x),\ \forall c\in\mathbb{R},\ \forall x\in[a,b]$

Together with these operations, the set $C[a,b]$ forms a vector space.

There is actually a great deal to say about vector spaces, but remembering that a human being is supposed to read this article, we will wrap up the subject without drawing it out too much. After all, our road leads to differential geometry. We will now add concepts such as inner products and norms to vector spaces. Left to my own devices, I would wander from here into Banach and Hilbert spaces, but time is limited, as you know: we need to discuss Euclid’s postulates and move on to differential geometry in the modern sense. Before I forget, I should mention that this series will not touch on anything involving complex numbers; consequently, all our definitions and theorems will concern real vector spaces. In complex vector spaces, conjugation enters the picture, and some properties may no longer hold. Since what we need for differential geometry is real vector spaces, however, those are the only ones we will discuss.

2. Inner Product Spaces

Readers who have heard of the dot product (scalar product) will already have some idea of what we are about to discuss. This concept is extremely important because it will allow us to measure the angle between two elements of a vector space, measure the length of a vector, and much more besides; we will use inner products throughout our work with curves and surfaces.

Definition 2. Let $(V,+,.)$ be a vector space. If the function

$$\langle\ ,\ \rangle:V\times V\to\mathbb{R}$$

defined on this vector space satisfies the following properties, it is called an inner product.

1. (Positive definiteness) $\langle u,u\rangle\ge 0,\ \forall u\in V$, with equality only when $u=0_V$. In other words, the inner product of any nonzero vector with itself is always positive, while the inner product of the zero vector with itself is zero.
2. We mentioned this before moving on to inner product spaces, but it is worth repeating here: let $V$ be a real vector space. Then, for all $u,v\in V$, the equality $\langle u,v\rangle=\langle v,u\rangle$ holds.
3. (Distributivity over addition) $\langle u+v,w\rangle=\langle u,w\rangle+\langle v,w\rangle,\ \forall u,v,w\in V$
4. (Compatibility with scalar multiplication) $\langle cu,v\rangle=c\langle u,v\rangle,\ c\in\mathbb{R},\ \forall u,v\in V$

Example 3. For the vector space $(\mathbb{R}^2,+,.)$, define the function $\langle\ ,\ \rangle:\mathbb{R}^2\times\mathbb{R}^2\to\mathbb{R}$ by

$$\left\langle \begin{bmatrix}u_1\\u_2\end{bmatrix}, \begin{bmatrix}v_1\\v_2\end{bmatrix} \right\rangle = u_1v_1+u_2v_2$$

This function is quite plainly an inner product; anyone unconvinced is welcome to prove it. (It can be shown very easily.)

If you are protesting, “But weren’t we going to measure angles between vectors? Weren’t we going to find their lengths?”, you are right: without the notion of a norm, the inner product does not have much of an application here. (I can already imagine the roasting I will get on Twitter. Fine, fine—it does have applications, just not for this particular purpose.)

Definition 3. Let $(V,+,.)$ be a vector space. If an inner product $\langle\ ,\ \rangle$ can be defined on this space, then $(V,\langle,\rangle)$ is called an inner product space.

Definition 4. Let $(V,+,.)$ be a vector space. A function $\|.\|:V\to\mathbb{R}$ defined on this space is called a norm.

1. For all $u\in V$, $\|u\|\ge 0$, with equality if and only if $u=0_V$.
2. For all $c\in\mathbb{R}$ and all $u\in V$, $\|cu\|=|c|\|u\|$.
3. (Triangle Inequality) For all $u,v\in V$, $\|u+v\|\le \|u\|+\|v\|$.

Example 4. For the vector space $(\mathbb{R}^2,+,.)$, define the function $\|.\|:\mathbb{R}^2\to\mathbb{R}$ by

$$\left\| \begin{bmatrix}u_1\\u_2\end{bmatrix} \right\| = \sqrt{u_1^2+u_2^2}$$

This function is a norm. The proof is quite straightforward.

Definition 5. Let $(V,+,.)$ be a vector space. If a norm $\|.\|$ can be defined on this space, then $(V,\|.\|)$ is called a normed space.

The example above should actually suggest an idea: the norm we defined looks rather like the square root of the vector’s inner product with itself. In other words, it seems that we can construct norms from inner products. And indeed we can.

Theorem 1. Let $(V,\langle,\rangle)$ be an inner product space. Define the function $\|.\|:V\to\mathbb{R}$ by

$$\|u\|=\sqrt{\langle u,u\rangle}$$

This function defines a norm; consequently, the space $(V,\|.\|)=(V,\sqrt{\langle,\rangle})$ is a normed space.

3. In the Next Chapter...

In this chapter, we briefly discussed vector spaces, inner product spaces, and normed spaces, all important tools for differential geometry. These are vast subjects about which dozens of books have been written. We covered only their most basic aspects—just enough to make differential geometry possible. Unfortunately, this chapter has no bibliography because we discussed very well-known concepts (we wrote the book, and now we are supposed to consult another book for the references???). In the next chapter, we plan to begin easing our way into Euclidean spaces.

Take very good care of yourselves!