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
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
A degree-$p$ calibration of an oriented Riemannian manifold $(X,g)$ is a differential p-form $\omega \in \Omega^p(X)$ with the property that
it is closed $d \omega = 0$;
evaluated on any oriented $p$-dimensional subspace of any tangent space of $X$, it is less than or equal to the induced degree-$p$ volume form, with equality for at least one choice of subspace.
A Riemannian manifold equipped with such a calibration is also called a calibrated geometry (Harvey-Lawson 82) or similar.
A calibrated submanifold of a manifold with calibration is an oriented submanifold such that restricted to each of its tangent spaces $\omega$ equals the induced volume form of the submanifold there.
Any calibrated submanifold $\Sigma \hookrightarrow X$ minimizes volume in its homology class.
For Let $\tilde \Sigma \hookrightarrow X$ be a homologous submanifold. Then Stokes theorem together with the condition that $d \phi = 0$ implies that the integration of differential forms of $\phi$ over $\Sigma$ equals that over $\tilde \Sigma$. The defining conditions on calibrations and on calibrated submanifolds then imply the inequality
under construction
For suitable $n$ and $p$, and given a real spin representation of $Spin(n)$, then the Cartesian space $\mathbb{R}^n$ with its canonical Riemannian structure becomes $p$-calibrated with the calibration form being
where
$\{e^a\}$ denotes the canonical linear basis of differential 1-forms;
$\epsilon$ is a non-vanishing spinor;
$\overline{\epsilon} \Gamma_{a_1 a_2 \cdots a_p} \epsilon$ is the canonical bilinear pairing which in components is given by evaluating $\epsilon$ in the quadratic form given by multiplying the skew-symmetrized product of $p$ of the representation matrices $\Gamma^a$ of the Clifford algebra with the charge conjugation matrix $C$.
(e.g. Dadok-Harvey 93).
For instance for $n = 7$ and $p = 3$ then this gives the associative 3-form calibration.
More generally for $X$ an $n$-dimensional Riemannian manifold with a covariantly constant spinor $\epsilon$, then under suitable conditions applying this construction in each tangent space gives a calibration.
The globalization of the associative 3-form of a G2-manifold is a calibration. A calibrated submanifold in this case is also called an associative submanifold.
The Cayley 4-form on Spin(7)-manifolds.
The original articles are
Reese Harvey, H. Blaine Lawson, Calibrated geometries, Acta Mathematica July 1982, Volume 148, Issue 1, pp 47-157
Reese Harvey, Calibrated geometries, Proceeding of the ICM 1983 (pdf)
The relation to Killing spinors goes back to
Reese Harvey, Spinors and Calibrations, Academic Press, 1990 (publisher)
Jiri Dadok, Reese Harvey, Calibrations and spinors, Acta Mathematica 1993, Volume 170, Issue 1, pp 83-120
See also
Wikipedia, Calibrated geometry
Jason Dean Lotay, Calibrated submanifolds and the Exceptional geometries, 2005 (pdf)
Discussion in string theory/M-theory includes the following.
Gary Gibbons, George Papadopoulos, Calibrations and Intersecting Branes (arXiv:hep-th/9803163)
Jerome Gauntlett, Neil Lambert, Peter West, Branes and Calibrated Geometries, Commun.Math.Phys. 202 (1999) 571-592 (arXiv:hep-th/9803216)
George Papadopoulos, Jan Gutowski, AdS Calibrations, Phys.Lett.B462:81-88,1999 (arXiv:hep-th/9902034)
Jan Gutowski, George Papadopoulos, Paul Townsend, Supersymmetry and generalized calibrations, Phys.Rev.D60:106006, 1999 (arXiv:hep-th/9905156)
Jan Gutowski, S. Ivanov, George Papadopoulos, Deformations of generalized calibrations and compact non-Kahler manifolds with vanishing first Chern class, Asian Journal of Mathematics 7 (2003), 39-80 (arXiv:0205012)
Last revised on April 2, 2019 at 13:33:49. See the history of this page for a list of all contributions to it.