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)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(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)
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
For $X$ a smooth manifold, a (parametrized oriented) smooth curve in $X$ is a smooth function $\gamma\colon \mathbb{R} \to X$ from the real line (or an interval therein) to $X$. (Compare path.)
For most purposes in differential geometry one needs to work with a regular curve, which is a parametrized smooth curve whose velocity, i.e. the derivative with respect to the parameter, is never zero. For example, this is important if one wants to split curve into segments which have no self-intersections, which is important.
In the foundations of differential topology, it is possible to define a tangent vector as an equivalence class of smooth curves at a given point in the image of the curve, effectively identifying a curve with its derivative at (say) $0$.
See also the fundamental theorem of differential geometry of curves?.
In a Cartesian space $\mathbb{R}^n$, an open smooth curve is a smooth curve $\gamma:\mathbb{R} \to \mathbb{R}^n$ which does not intersect itself: For every real number $a \in \mathbb{R}$ and $b \in \mathbb{R}$, the distance between the two points on the curve parameterized by $a$ and $b$ is greater than zero: $\rho(\gamma(a),\gamma(b)) \gt 0$. Equivalently, an open smooth curve is a smooth curve such that the shape of the image of $\gamma$ is contractible: $\esh \mathrm{im}(\gamma) \simeq *$.
A closed smooth curve is a smooth curve $\gamma:\mathbb{R} \to \mathbb{R}^n$ which does intersect itself: for every real number $a \in \mathbb{R}$, there is a real number $b \in \mathbb{R}$, such that $\gamma(a) = \gamma(a + b)$. Equivalently, a closed smooth curve is a smooth curve such that the shape of the image of $\gamma$ is equivalent to the circle type: $\esh \mathrm{im}(\gamma) \simeq S^1$.
In classical mathematics, every smooth curve $\gamma:\mathbb{R} \to \mathbb{R}^n$ is either open or closed. In constructive mathematics, there are smooth curves where it cannot be proved to be either open or closed, due to the failure of trichotomy.
In algebraic geometry, an algebraic curve is a $1$-dimensional algebraic variety over a field.
An example: elliptic curve.
See also
Examples of sequences of local structures
geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|
$\leftarrow$ differentiation | integration $\to$ | |||||||
smooth functions | derivative | Taylor series | germ | smooth function | ||||
curve (path) | tangent vector | jet | germ of curve | curve | ||||
smooth space | infinitesimal neighbourhood | formal neighbourhood | germ of a space | open neighbourhood | ||||
function algebra | square-0 ring extension | nilpotent ring extension/formal completion | ring extension | |||||
arithmetic geometry | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ integers | ||||
Lie theory | Lie algebra | formal group | local Lie group | Lie group | ||||
symplectic geometry | Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |
Last revised on May 18, 2022 at 18:28:29. See the history of this page for a list of all contributions to it.