manifolds and cobordisms
cobordism theory, Introduction
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
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differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
For an embedding of manifolds, a tubular neighbourhood of in is
a real vector bundle ;
an extension of to an isomorphism
with an open neighbourhood of in .
The derivative of provides an isomorphism of with the normal bundle of in .
For instance (DaSilva, theorem 3.1).
Moreover, tubular neighbourhoods are unique up to homotopy in a suitable sense:
For an embedding , write for the topological space whose underlying set is the set of tubular neighbourhoods of and whose topology is the subspace topology of equipped with the C-infinity topology.
If and are compact manifolds, then is contractible for all embeddings .
This appears as (Godin, prop. 31).
(…) propagating flow (…) (Godin).
key application: Pontrjagin-Thom collapse map
Basics on tubular neighbourhoods are for instance in section 3 of
The homotopical uniqueness of tubular neighbourhoods is discussed in
For an analogue in homotopical algebraic geometry see
See also
Last revised on February 8, 2019 at 08:22:29. See the history of this page for a list of all contributions to it.