8 - Metric notions

Summary

One may say that if there is no metric, there is no geometry. In the present synthetic context, the notion of Riemannian (and pseudo-Riemannian) metric comes, for manifolds, in a combinatorial manifestation, besides in the classical manifestation in terms of the tangent bundle. We shall utilize both manifestations, and their interplay. The combinatorial notion deals essentially with points which are second-order neighbours, x ∼₂y.

Pseudo-Riemannian metrics

Both the combinatorial and the classical notion are subordinate to the notion of quadratic differential form, which likewise comes in a combinatorial and in a classical manifestation. A Riemannian metric on M will be a quadratic differential form with a certain positive-definiteness property.

We begin with the combinatorial notions (cf. Kock, 1998).

Definition 8.1.1A (combinatorial) quadratic differential form on a manifold M is amp g: M₍₂₎ → R vanishing on M₍₁₎ ⊆ M₍₂₎.

Note the analogy with the notion of differential R-valued 1-form on M, which is (cf. Definition 2.2.3) a map M₍₁₎ → R vanishing on M₍₀₎ ⊆ M₍₁₎. (Recall that M₍₁₎ is the diagonal in M×M.)

The canonical example is M = Rⁿ, with

whose meaning is the square distance between and. (The distance itself cannot well be formulated for neighbour points (of any order), it seems.)

Quadratic differential forms are always symmetric:

Proposition 8.1.2Let g: M₍₂₎ → R be a quadratic differential form. Then g is symmetric, g(x, y) = g(y, x), for any x ∼₂y. Furthermore, g extends uniquely to a symmetric M₍₃₎ → R.

Proof. The assertions are local, so we may assume that M is an open subset of a finite-dimensional vector space V.