#019: Show equivalence of "inner products"/bilinear forms and trigonometry

Category: Linear algebra - Tags: Riemannian geometry, Pseudo-Riemannian geometry, symplectic geometry

Show the equivalence of a linear space with a positive-definite symmetric bilinear form and geometry with the law of cosines (and suitable adjustment if the form is not positive definite, but still non-degenerate); show the equivalence of a linear space with a skew-symmetric form and geometry with areas.

Mathematical problem. To really bridge linear algebra and geometry, we are looking for a connection such that the definition on each side are enough to recover the other. There are three cases we want to cover:

1. positive-definite symmetric bilinear form - this defines lengths and angles between segments (represented by vectors). The connection is the law of cosines by comparing $$c^2 = a^2 + b^2 - 2ab \cos \alpha$$ with $$c \cdot c = (a - b) \cdot (a - b) = a \cdot a + b \cdot b - 2 a \cdot b$$.
2. non-degenerate symmetric bilinear form - note that, for orthogonal components, the law of cosine becomes the Pythagorean theorem $$c^2 = a^2 + b^2$$. If one of the direction “negative-definite” we have $$c^2 = a^2 - b^2$$. This is still the Pythagorean theorem, except that $$a$$ and $$b$$ no longer represent orthogonal components, but $$c$$ and $$b$$ do. The idea, then, is to re-express the law of cosines and find the proper connection between vectors and geometry.
3. non-degenaret skew-symmetric bilinear form - this defines only areas of parallelopipeds. The connection should simply be that the area is $$ab \sin \alpha$$. Note that the sine is an odd function, therefore if we set $$\omega(a,b) = |a||b| \sin \alpha$$ the form must be skew-symmetric.

Physical significance. We want to make a tight connection between the three different types of differential geometries used in physics, so that we can have a crips geometrical interpretation for the physics. The first case is used in standard physical space. The second case is used in space-time. The third case is used in phase space. The important aspect here is showing that the assumptions on the existence of a specific form with specific properties are equivalent to the geometrical (and therefore physical) assumptions.

Another issue is that, mathematically, these are usually treated as different subjects. We believe is useful to see them as variation of a general theme, so that one can build a more general intuition for the tools that are common (i.e. all of differential topology, linear algebra, …).