Geometry of bracket-generating distributions of step 2 on graded manifolds
Abstract
A $Z_2-$graded analogue of bracket-generating distribution is given. Let $\cd$ be a distribution of rank $(p,q)$ on an $(m,n)$-dimensional graded manifold $\cm,$ we attach to $\cd$ a linear map $F$ on $\cd$ defined by the Lie bracket of graded vector fields of the sections of $\cd.$ Then $\mathcal{D}$ is a bracket-generating distribution of step $2$, if and only if $F$ is of constant rank $(m-p, n-q)$ on $\cm$.
Keywords
References
- [1] A. Bejancu, On bracket-generating distributions, Int. Electron. J. Geom. 3 (2010) no. 2, 102 - 107.
- [2] O. Goertsches, Riemannian supergeometry, Math. Z., 260 (2008) 557-–593.
- [3] J. Monterde and J. Munoz-Masque and O. A. Sanchez-Valenzuela, Geometric properties of involutive distributions on graded manifolds, Indag. Mathem., N.S., 8 (1997), 217-246.
- [4] S. Vacaru and H. Dehnen, Locally Anisotropic Structures and Nonlinear Connections in Einstein and Gauge Gravity, Gen. Rel. Grav., 35 (2003) 209-250.
- [5] S. I. Vacaru, Superstrings in higher order extensions of Finsler Superspaces, Nucl. Phys. B 494 (1997) no. 3, 590-656.
- [6] V. S. Varadarajan, Supersymmetry for mathematicians: an introduction, Courant Lecture Notes Series, New York, 2004.
- [7] P. C. West, Introduction to supersymmetry and supergravity, Second Edition, World Scientific Pub Co Inc, 1990.
- [8] C. D. Zanet, Generic one-step bracket-generating distributions of rank four, Archivum Mathematicum, 51 (2015), 257 - 264.
