The 4 dimensional spacetime manifold is a typical example of pseudo-Riemannian manifold. Are there other mathematically or physically interesting example of it?

  • $\begingroup$ It isn't quite what you're asking, but one of the slickest constructions of $n$-dimensional hyperbolic space $\mathbb{H}^n$ (i.e., the hyperboloid model) is as a space-like hypersurface in $(n+1)$-dimensional Minkowski space. $\endgroup$ – Branimir Ćaćić Jul 31 '18 at 8:06

A Lie group $G$ acts on itself both by left translation (group multiplication on the left) and by right translation (group multiplication on the right). Hence, it's natural to look for Riemannian metrics that are bi-invariant, i.e., that make both left translation and right translation by a group element into isometries. However, there's a folkloric result that $G$ admits a bi-invariant Riemannian metric if and only if $G \cong H \times K$ for $H$ compact and $K$ abelian, which means, for instance, that the non-compact simple Lie group $$ SL(2,\mathbb{R}) := \{A \in M_2(\mathbb{R}) \mid \det(A) = 1\} $$ does not admit any bi-invariant Riemannian metrics. However, as we'll see below, it does admit a canonical bi-invariant pseudo-Riemannian metric of signature $(+,-,-)$.

Now, in general, if $\mathfrak{g}$ is the Lie algebra of $G$, we can define the Killing form $B : \mathfrak{g} \times \mathfrak{g} \to \mathbb{R}$ by $$ \forall X, Y \in \mathfrak{g}, \quad B(X,Y) := -\operatorname{trace}(\operatorname{ad}(X)\operatorname{ad}(Y)), $$ where $\operatorname{ad}(X) : \mathfrak{g} \to \mathfrak{g}$ is defined by $\operatorname{ad}(X)Y := [X,Y]$. The Killing form $B$ is bilinear, symmetric, and invariant under the adjoint action of $G$ on $\mathfrak{g}$, so by the identification of $\mathfrak{g}$ with left-invariant vector fields on $G$, it induces a bi-invariant possibly degenerate semi-Riemannian metric on $G$ with the same signature as $B$. If $G$ is semi-simple, then $B$ is actually non-degenerate, and hence induces a completely canonical bi-invariant pseudo-Riemannian metric; moreover, the Killing form $B$ is positive-definite (and hence yields a bi-invariant Riemannian metric) if and only if $G$ is compact. So, for instance, the compact semi-simple Lie group $$ SU(2) := \{A \in M_2(\mathbb{C}) \mid A^\ast A = I_2, \ \det(A) = 1\} $$ admits a canonical bi-invariant Riemannian metric induced by its Killing form, but the non-compact simple (and hence, semi-simple) Lie group $SL(2,\mathbb{R})$, which admits no bi-invariant Riemannian metric, nonetheless admits a canonical pseudo-Riemannian metric of signature $(+,-,-)$ induced by its Killing form.

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