Eleven-Dimensional Oriented Lorentzian Geometry

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, as in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). An oriented Lorentzian eleven-manifold is an oriented smooth manifold $M$ of dimension $11$, without boundary, together with a smooth symmetric covariant two-tensor $g$ whose inertia at every point is $(10,1,0)$: ten positive and one negative eigenvalue and no zero eigenvalues (Smooth manifolds and their smooth charts, Oriented real bundles and oriented frame bundles, Positive and negative definiteness, the inertia $(p,q,r)$, rank $p+q$, and signature $p-q$ of a real symmetric bilinear or quadratic form). The metric is therefore nondegenerate and has signature $(-+\cdots +)$. Its inverse is denoted $g^{-1}$.

For $0\le p\le 11$, let ${\mathrm{\Omega }}^p(M)$ be the smooth sections of ${\mathrm{\Lambda }}^p{T}^{\ast }M$ (A smooth differential $k$-form). The metric induces the signed pairing

$⟨\alpha ,\beta {⟩}_g=\frac{1}{p!}{\alpha }_{M_1\cdots M_p}{\beta }^{M_1\cdots M_p},$

where all indices are raised with $g^{-1}$; set $\mathrm{\mid }\alpha {\mathrm{\mid }}_g^2:=⟨\alpha ,\alpha {⟩}_g$. This pairing need not be positive. Let ${\mathrm{vol}}_g$ be the unique smooth eleven-form positive in the chosen orientation and locally equal to $\sqrt{\mathrm{\mid }\det (g_{MN})\mathrm{\mid }}d{x}^0\wedge \cdots \wedge d{x}^{10}$ in positively oriented coordinates. The Lorentzian Hodge star ${\ast }_g:{\mathrm{\Omega }}^p(M)\to {\mathrm{\Omega }}^{11-p}(M)$ is determined pointwise by

$\alpha \wedge {\ast }_g\beta =⟨\alpha ,\beta {⟩}_g{\mathrm{vol}}_g(\alpha ,\beta \in {\mathrm{\Omega }}^p(M))\mathrm{.}$

This map exists uniquely: in a basis of increasing coordinate wedges, each $p$-wedge pairs with its complementary $(11-p)$-wedge with coefficient $+1$ or $-1$, so the wedge pairing is perfect. The metric-induced pairing on ${\mathrm{\Lambda }}^p{T}_x^{\ast }M$ is nondegenerate because it is the exterior-power pairing induced by the invertible map $g^{-1}$. Thus for each ${\beta }_x$ there is exactly one $({\ast }_g\beta {)}_x$ with the displayed values against every ${\alpha }_x$. In a coordinate frame the coefficients depend smoothly on $g$ and $g^{-1}$; matrix inversion preserves smoothness for nonzero determinant (Matrix inversion preserves $C^k$ regularity where the determinant is nonzero), so these pointwise maps form a smooth bundle map. The convention for differential forms and wedge products is that of A smooth differential $k$-form and The wedge product of differential forms.

This is a mathematical structure definition. It asserts neither the existence of a metric on every eleven-manifold nor the existence of a spin structure.