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
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
For $(X,g)$ a Riemannian manifold and $p \in X$ a point, the geodesic flow at $p$ is the map defined on an open neighbourhood of the origin in $(T_p X ) \times \mathbb{R}$ that sends $(v,r)$ to the endpoint of the geodesic that starts with tangent vector $v$ at $p$ and has length $r$.
(…)
Let $(X,g)$ be a Riemannian manifold
(…) geodesic flow (…)
The following are some auxiliary definitions that serve to analyse properties of geodesic flow (see Properties).
For $p \in X$ a point and $r \in \mathbb{R}$ a positive real number, we write
for set of points which are of distance less than $r$ away from $p$. As the propositions below assert, for small enough $r$ this is diffeomorphic to an open ball and we speak of metric balls or geodesic balls .
For $p \in P$ a point, the injectivity radius $inj_p \in \mathbb{R}$ is the supremum over all values of $r \in \mathbb{R}$ such that the geodesic flow starting at $p$ with radius $r$ $\exp(-) : B_r(T_p X) \to X$ is a diffeomorphism onto its image.
The injectivity radius of $(X,g)$ is the infimum of the injectivity radii at each point.
The injectivity radius is
either equal to half the length of the smalled periodic geodesic,
or equal to the smallest distance between two conjugate points.
This appears for instance as scholium 91 in (Berger).
There are several lower boundas on the injectivity radius of a Riemannian manifold.
The convexity radius is always less than or equal to half of the injectivity radius:
This appears for instance as proposition IX.6.1 in Chavel, where it is attributed to M. Berger (1976). In (Berger) it is proposition 95.
Let $R$ be the Riemann curvature tensor of $g$. For $p \in X$ the sectional curvature of a plane spanned by vectors $v,w \in T_p X$ is
Say that $(X,g)$ is complete if, equivalently,
with the distance function $X$ is a complete metric space;
$(X,g)$ is geodesically complete in that for all $v \in T_p X$ the flow $t \mapsto \exp_p(t v)$ exists for all $t \in \mathbb{R}$.
Let $(X,g)$ be complete and such that
the absolute value of the sectional curvature at all points is bounded from above;
the volume of the geodesic unit ball at all points is bounded from below.
Then the injectivity radius is positive.
This is due to (CheegerGromovTaylor). A survey is in (Grant).
Every paracompact manifold admits a complete Riemannian metric with
bounded absolute sectional curvature;
positive convexity radius
and hence with positive injectivity radius.
This is shown in (Greene).
The following is literature on injectivity radius estimates
A general exposition is in sectin 6 “Injectivity, Convexity radius and cut locuss” of
Also section IX of
A survey of the main estimates is in
The main theorem is due to
Older results on compact manifolds are in
The existence of metrics with all the required propertiers for the injectivity estimates (completeness, bounded absolute sectional curvature) on paracompact manifolds is shown in
More discussion of construction of Riemannian manifolds with bounds on curvature and volume is in
Analogous results for Lorentzian manifolds are discussed in