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
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
\array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }
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Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
formal smooth ∞-groupoid?
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.
Last revised on February 27, 2015 at 14:57:03. See the history of this page for a list of all contributions to it.