Need introduction to coordinate-free geometry

Question

Forgive my whiny tone -- I'm driving myself nuts. These questions have been driving me crazy and I'm losing objectivity. I need a serious introduction to coordinate-free geometry. Let me ask more than one question, not because I deserve multiple answers but because I want to show exactly how I am confused and thus get a better reference. I'm after a reference, not answers.

I've searched Amazon's book section and it returns books that are pretty obviously the stuff I studied back in the 70s. A google search for coordinate-free geometry tutorial tends to return stuff that breezes past all the elementary stuff and launches into differential geometry and so on. I'm not ready for that.

This was provoked by chapter 1 of Thorne and Blandford's book, wherein the dot product is defined by

$$A \cdot B = \frac{1}{4}[(A+B)^2 - (A-B)^2]$$

Now, I can expand the rhs with the best of them.

$$A \cdot B = \frac{1}{2}[AB + BA]$$

But, what, precisely, is $AB$ in the above? This is the kind of stuff I'm having trouble with. Where can I get a real introduction to coord-free geometry that will show me the algebra?

Allow me to drone on a bit. Since I'm having trouble getting graphics into my latex doc let me play a little fast and loose with my example. Consider the unit circle at the origin. There's a tangent line to that circle which is horizontal and runs through (0,1). That's what I think of when I hear the word tangent.

Aside: if I want to bend a sheet of aluminum I can take a knife and a straightedge and score a line on the aluminum and then bend it. OTOH, as Lincoln said, ``four score and seven years ago....'' Now, that's one word, score, used for two completely separate things.

Back to my question: If I try to read a book on manifolds I tend to run into tangent space. I can't get anyone to say that tangent space doesn't mean tangent at all. If we "reserve" tangent to mean straight lines sitting on curves, we don't "lose" anything by calling tangent space something like George space. That is, there's nothing tangent about tangent space. Right?

From the descriptions of tangent space it appears that what it really boils down to is the set of all vectors using the particular point as the origination point. That is, that's what I'm getting out of it. Ah, here's another way to get at my issue: I'm worried that there's something I'm missing in the descriptions of tangent space that somehow includes some facet of tangent.

(Sigh.) One more thing occurs to me: Two points, $P$ and $Q$. Deliberately ignoring/omitting the vectors between $P$ and $Q$, assuming I'm right about what a tangent space is, that means that a vector originating at $Q$ is not part of $P$'s tangent space. Am I right?

(Deeper sigh.) Also, I need a proper definition of this operator: $\otimes$.

Answer 1

IMHO, the definitions by Thorne and Blandford is needlessly confusing (beside lacking rigor)

A vector space is a set (whose elements we call vectors) whose elements we can

• add together
• multiply by scalars (in physics usually real or complex numbers).

We also require that these operations have "nice" properties, see here for more details.

An example of a vector space is the set of arrows with a common base point in the euclidean plane or 3D space, but there are many others.

A vector space can then be equiped with a dot product, which is a bilinear map $(A,B) \mapsto A\cdot B$ such that :

• $A\cdot B = B\cdot A$ for every two vectors $A$ and $B$ (we say that it is symmetric)
• $A\cdot A >0$ for any non-zero vector $A$

If we are given a dot product, we can define the length of a vector $A$ as $\|A\| =\sqrt{A\cdot A}$. Then, we can derive the so-called polarization identity : $$A\cdot B = \frac{1}{4}\Big(\|A+B\|^2 - \|A-B\|^2\Big)$$

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Proof : Let $A,B$ be two elements of a vector space equipped with a dot product : \begin{align} \|A+B\|^2 &= (A+B)\cdot(A+B) \\ &= A \cdot(A+B) + B\cdot(A+B) \\ &= A\cdot A + A\cdot B + B\cdot A + B\cdot B \\ &= \|A\|^2 + 2A \cdot B + \|B\|^2 \end{align} similarly we find :

\begin{align} \|A-B\|^2 &= \|A\|^2 - 2A\cdot B + \|B\|^2 \end{align}

Therefore, we have : \begin{align} \frac14(\|A+B\|^2-\|A-B\|^2)&= \frac14\big(\|A\|^2 + 2A \cdot B + \|B\|^2-(\|A\|^2 + 2A \cdot B + \|B\|^2\big)\\ &= A\cdot B \end{align}

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To learn more about these points, take a book on linear algebra (some are recommended here).

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About the definition of tangent space : the use of the same word to describe different things need not be a problem, as context will usually make it clear which meaning is intended. When some of the different definition occur in the same context, it is best if they agree. Here, there is actually no problem, since there is the tangent, the tangent space and tangent vectors, so no confusion is possible.

Further more :

• If you consider a manifold immersed in euclidean space and a curve running in that manifold, than the tangent vector of the curve at a point will lie in the tangent space of the manifold at that point. In fact, the tangent space of the manifold $M$ at a point $p$ is exactly the set of tangent vectors to curves in $M$ running through $p$.

• A regular smooth curve $\gamma$ in euclidean affine space $A$ is an immersed submanifold. The tangent to $\gamma$ at a point $p$ is an affine subspace of $A$, whose direction is exactly the tangent space of $\gamma$ at $p$.

So actually, the different uses of tangent actually match pretty well.

Answer 2

Your product AB is called the geometric product, and it's what defines vectors in the first place. A dot product is the symmetric part of that basic product.