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The fundamental theorem of affine/projective geometry says that a bijection between two finite dimensional spaces that preserves the relation of collinearity is a (semi-) affine/projective isomorphism.

Fundamental Theorem of Affine geometry let $X,X'$ be two finite dimensional affine spaces over two fields $K,K'$ of same dimension $d\geq 2$, and let $f:X\to X'$ be a bijection that sends collinear points to collinear points, i.e. such that for all $a,b,c\in X$ that are collinear, $f(a),f(b),f(c)$ are collinear too. Then $f$ is a semi-affine isomorphism.

This means that there is a field isomorphism $\sigma:K\to K'$ such that for any point $a\in X$ the map induced by $f_a: X_a\to X'_{f(a)}$ is a $\sigma$-semi-linear isomorphism.

Fundamental Theorem of Projective geometry let $P(X),P(X')$ be two finite dimensional projective spaces over two fields $K,K'$ of same dimension $d\geq 2$, and let $f:P(X)\to P(X')$ be a bijection that sends collinear points to collinear points. Then $f$ is a semi-linear isomorphism.

What are some applications of this? I remember that you can use the projective verion of this to prove that any automorphism of $SO(3,\Bbb R)$ is given by conjugation with some orthogonal matrix. Are there some other beautiful applications?

In the time since I asked this question I have tried to find some applications of this. If one wishes to apply this theorem one needs to be in a situation where there naturally arise lines. I thought of two such situations:

  • Non-degenerate quadratic forms on a $2$ dimensional vector space over an algebraically closed field, always have two isotropic lines. Are there more or less natural things one can do to these forms that aren't obviously linear (or affine) that could be shown to be affine when looking how the isotropic lines vary?
  • There is a classical exercise where one shows that endomorphisms $\phi$ of the vector space of $k$-linear maps $\mathrm{End}_k(V)$ that preserve rank are given by left and right composition with isomorphisms of the vector space $V$ (or something similar where there is also transposition). One looks at a standard basis of rank one operators and shows that either all rank one operators with given kernel are sent to rank one operators with the same kernel or are sent to rank one operators with the same image. In any case there is a bijection from the rank one operators on $V$ to $P(V)\times P(V^*)$ given by $u\mapsto (\mathrm{Im}(u),\mathrm{Ker}(u))$, and this, when composed with the rank preserving endomorphism $\phi$ should send collinear points in $\pi(\lbrace$rank $1$ operators$\rbrace)\subset P(\mathrm{End}_k(V))$ to pairs of collinear points. One should be able to get a pair of maps $f:P(V)\to P(V)$ and $g:P(V^*)\to P(V^*)$ (or from th projective spaces to their dual projectve spaces) that preserve collinearity, which would yield the result.

I would very much like to learn substantial applications of the fundamental theorems, so I set a bounty on this question.

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1 Answer 1

One of the fundamental theorems in quantum mechanics is Wigner's theorem. It says that a map preserving the inner product on a complex Hilbert space is unitary or anti-unitary. It's a fairly easy consequence of the generalization of the fundamental theorem of projective geometry to infinite-dimensional spaces, which itself follows easily from the theorem as you stated it.

From the perspective of the mathematical foundations of quantum mechanics, it shows that observables have to correspond to unitary or anti-unitary operators. A map as above is called a symmetry and the citation for Wigner's 1963 Nobel prize in Physics included the phrase "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles", see www.nobelprize.org/nobel_prizes/physics/laureates/1963/

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