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Cayley fibrations in the Bryant-Salamon $Spin(7)$ manifold

Federico Trinca

Abstract

On each complete asymptotically conical $Spin(7)$ manifold constructed by Bryant and Salamon, including the asymptotic cones, we consider a natural family of $SU(2)$ actions preserving the Cayley form. For each element of this family, we study the (possibly singular) invariant Cayley fibration, which we describe explicitly, if possible. These can be reckoned as generalizations of the trivial flat fibration of $\mathbb{R}^8$ and the product of a line with the Harvey-Lawson coassociative fibration of $\mathbb{R}^7$. The fibres will provide new examples of asymptotically conical Cayley submanifolds in the Bryant-Salamon manifolds of topology $\mathbb{R}^4, \mathbb{R}\times S^3$ and $\mathcal{O}_{CP^1} (-1)$.

Cayley fibrations in the Bryant-Salamon $Spin(7)$ manifold

Abstract

On each complete asymptotically conical manifold constructed by Bryant and Salamon, including the asymptotic cones, we consider a natural family of actions preserving the Cayley form. For each element of this family, we study the (possibly singular) invariant Cayley fibration, which we describe explicitly, if possible. These can be reckoned as generalizations of the trivial flat fibration of and the product of a line with the Harvey-Lawson coassociative fibration of . The fibres will provide new examples of asymptotically conical Cayley submanifolds in the Bryant-Salamon manifolds of topology and .

Paper Structure

This paper contains 29 sections, 27 theorems, 129 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spin(7) manifolds
  4. Cayley submanifolds and Cayley fibrations
  5. Riemannian conifolds
  6. Bryant--Salamon Lg manifolds
  7. The negative spinor bundle of Lg
  8. The Lg-structures
  9. Automorphism Group
  10. The Cayley fibration invariant under the Lg action on the fibre
  11. The Cayley fibration invariant under the lift of the Lg action on Lg
  12. The choice of coframe on Lg
  13. The horizontal and the vertical space
  14. The Lg action
  15. Lg adapted coordinates
  16. ...and 14 more sections

Key Result

Proposition \oldthetheorem

The subspace spanned by tangent vectors $u,v,w,y$ is a Cayley 4-plane, up to orientation, if and only if the following form vanishes: where and

Figures (7)

  • Figure 1: Level sets of $F$ in the generic and in the conical case
  • Figure 2: Approximation of a Cayley at $u=0$ when $\alpha_0\in(0,\pi/2)$
  • Figure 3: Approximation of a Cayley at $u=0$ when $\alpha_0=\pi/2$
  • Figure 4: Level sets of the multi-moment map in the generic and conical case
  • Figure 5: Flow lines for $(\ref{['final ODE Sp(1)xId_1']})$.
  • ...and 2 more figures

Theorems & Definitions (76)

  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Proposition \oldthetheorem: Karigiannis--Min-Oo KM05
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • ...and 66 more