synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
The notion of pseudo-Riemannian metric is a slight variant of that of Riemannian metric.
Where a Riemannian metric is governed by a positive-definite bilinear form, a Pseudo-Riemannian metric is governed by an indefinite bilinear form.
This is equivalently the Cartan geometry modeled on the inclusion of a Lorentz group into a Poincaré group.
A pseudo-Riemannian “metric” is a nondegenerate quadratic form on a real vector space $\mathbb{R}^n$. A Riemannian metric is a positive-definite quadratic form on a real vector space. The data of such a quadratic form may be equivalently given by a nondegenerate symmetric bilinear pairing $\langle\, , \, \rangle$ on $\mathbb{R}^n$.
A pseudo-Riemannian metric
can always be diagonalized: there exists a basis $e_1, \ldots, e_n$ such that
where the pair $(p, n-p)$ is called the signature of the form $Q$. Pseudo-Riemannian metrics on $\mathbb{R}^n$ are classified by their signatures; thus we have a standard metric of signature $(p, n-p)$ where $\{e_1, \ldots, e_n\}$ is the standard basis of $\mathbb{R}^n$.
More generally, there is a notion of pseudo-Riemannian manifold (of type $(p, n-p)$, which is an $n$-dimensional manifold $M$ equipped with a global section
of the bundle of symmetric bilinear forms over $M$, such that each $\sigma(x)$ is a nondegenerate form on the tangent space $T_x(M)$.
Certain theorems of Riemannian geometry carry over to the more general pseudo-Riemannian setting; for example, pseudo-Riemannian manifolds admit Levi-Civita connections, or in other words a unique notion of covariant differentiation of vector fields
such that
In that case, one may define a notion of geodesic in pseudo-Riemannian manifolds $M$, and we have a notion of “distance squared” between the endpoints along any geodesic path $\alpha: [0, 1] \to M$ (which might be a negative number of course). The term “pseudo-Riemannian metric” may refer to such distances in general pseudo-Riemannian manifolds. (I guess.)
A typical example of pseudo-Riemannian manifold is a Lorentzian manifold, where the metric is of type $(1, n-1)$. This is particularly so in the case $n = 4$, where such manifolds are the mathematical backdrop for studying general relativity and cosmological models.