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
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
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)
On a Riemannian manifold $(X,g)$, a geodesic (or geodesic line, geodesic path) is a path $x : I \to X$, for some (possibly infinite) interval $I$, which locally minimizes distance.
One way to define a geodesic is as a critical point of the functional of length
on the appropriate space of curves $I \to X$, where $g$ is the metric tensor. This implies that it satisfies the corresponding Euler-Lagrange equations, which in this case means that the covariant derivative (for the Levi-Civita connection)
vanishes.
In local coordinates, with Christoffel symbols $\Gamma^i_{jk}$ the Euler-Lagrange equations for geodesics form a system
So this means that a curve is a geodesic if at every point its tangent vector is the parallel transport of the tangent vector at the start point along the curve.
A geodesic may not globally minimize the distance between its end points. For instance, on a 2-dimensional sphere, geodesics are arcs of great circle?s. Any two distinct non-antipodal points are connected by exactly two such geodesics, one shorter than the other (you can go from Los Angeles to Boston directly across North America, or the long way around the world).
A geodesic which does globally minimize distance between its end points is called a minimizing geodesic. The length of a minimizing geodesic between two points defines a distance function for any Riemannian manifold which makes it into a metric space.
en.wikipedia: geodesic
Springer eom: Yu. A. Volkov, Geodesic line; Yu. A. Volkov, Geodesic coordinates; Geodesic distance; V.A. Zalgaller, Geodesic geometry; D.V. Anosov, Geodesic flow
Sh. Kobayashi, K. Nomidzu, Foundations of differential geometry, vol 1, 1963, vol 2, 1969, Wiley Interscience; reedition 1996 in series Wiley Classics; Russian ed.: Nauka, Moscow 1981.
Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 10, Springer-Verlag 1987, xii + 510 pp. (for a review see Bull. AMS and MR88f:53087); reprinted 2008, Springer Classics in Math.