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
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Given a tangent vector field on a differentiable manifold then its flow is the group of diffeomorphisms of that lets the points of the manifold “flow along the vector field” hence which sends them along flow lines (integral curves) that are tangent to the vector field.
Integral curves generalize to integral sections for multivector fields. See there for more.
Throughout, let be a differentiable manifold and let be a continuously differentiable vector field on (i.e. of class ).
(integral curves/flow lines)
An integral curve or flow line of the vector field is a differentiable function of the form
for an open interval with the property that its tangent vector at any equals the value of the vector field at the point :
(flow of a vector field)
A global flow of is a function of the form
such that for each the function is an integral curve of (def. ).
A flow domain is an open subset such that for all the intersection is an open interval containing .
A flow of on a flow domain is a differentiable function
such that for all the function is an integral curve of (def. ).
(complete vector field)
The vector field is called a complete vector field if it admits a global flow (def. ).
In synthetic differential geometry a tangent vector field is a morphism such that
The internal hom-adjunct of such a morphism is of the form
If is sufficiently nice (a microlinear space should be sufficient) then this morphism factors through the internal automorphism group inside the internal endomorphisms
Then a group homomorphism
with the property that restricted along any of the affine inclusions it equals
is a flow for .
Let be a global flow of a vector field (def. ). This yields an action of the additive group of real numbers on the differentiable manifold by diffeomorphisms, in that
;
;
.
(fundamental theorem of flows)
Let be a smooth manifold and a smooth vector field. Then has a unique maximal flow (def. ).
This unique flow is often denoted or (see also at exponential map).
e.g. Lee, theorem 12.9
Let be a compact smooth manifold. Then every smooth vector field is a complete vector field (def. ) hence has a global flow (def. ).
e.g. Lee, theorem 12.12
Last revised on April 18, 2024 at 16:47:45. See the history of this page for a list of all contributions to it.