nLab twistor fibration

Idea

The twistor fibration $\mathbb{C}P^3 \to S^4$ (Atyiah-Hitchin-Singer 78, Ex. 2 on p. 438, Atiyah 79, Sec. III.1, see also Bryant 82, ArmstrongSalamon14, ABS 19), also called, in its coset space-version $SO(5)/U(2) \to SO(5)/SO(4)$, the Calabi-Penrose fibration (apparently starting with Lawson 85, Sec. 3, see also, e.g., Loo 89, Seade-Verjovsky 03, 3 for this usage, see Nordstrom 08, Lemma 2.31 for review of Calabi’s construction Calabi 67, Calabi 68) is a fiber bundle-structure on complex projective 3-space over the 4-sphere with 2-sphere (Riemann sphere) fibers:

$$\array{ \mathbb{C}P^1 &\longrightarrow& \mathbb{C}P^3 \\ && \big\downarrow^{\mathrlap{p}} \\ && S^4 }$$

If one identifies the 4-sphere as the quaternionic projective line $S^4 \simeq \mathbb{H}P^1$, then the fibration $p$ here is given by sending complex lines to the quaternionic lines which they span (Atiyah 79, III (1.1), see also Seade-Verjovsky 03, p. 198):

$$\array{ \mathbb{C}P^3 &\overset{p}{\longrightarrow}& \mathbb{H}P^1 \\ \{x \cdot z \vert z \in \mathbb{C}\} &\mapsto& \{x \cdot q \vert q \in \mathbb{H}\} } \,,$$

for any $x \in \mathbb{C}^4 \simeq_{\mathbb{R}} \mathbb{H}^2$.

Generalizations

It is possible to define a twistor fibration over each $S^{2n}$, where the resulting manifold is a complex manifold endowed with a holomorphic? $n$-plane field transverse to the fibers of the mapping. Namely, writing $S^{2n}=SO(2n+1)/SO(2n)$, then, using the inclusion $U(n) \subset SO(2n)$, one has the coset fibration

$$Z_n=SO(2n+1)/U(n) \to SO(2n+1)/SO(2n).$$

The manifold $Z_n$ canonically has the structure of a complex manifold and is known as the twistor space of $S^{2n}$.

There are generalizations of this picture for each of the so-called ‘inner’ symmetric spaces $G/K$ where $K$ is the fixed subgroup of an involution that is an inner automorphism of $G$. The twistor fibration is of the form $G/U \to G/K$ where $U \subset K$ is a subgroup such that $K/U$ (the typical fiber of the fibration) is an Hermitian symmetric space. There are also other kinds of twistor spaces over $G/K$ that are flag manifolds of the form $G/T$ where $T \subset K$ is a maximal torus. (For these generalizations see Bryant 85.)

Related concepts

• quaternionic Hopf fibration

• twistor space, complex projective 3-space

• metric cone over complex projective 3-space

References

• Eugenio Calabi, Minimal immersions of surfaces in euclidean spheres, J. Differential Geometry (1967), 111-125 (euclid:jdg/1214427884)

• Eugenio Calabi, Quelques applications de l’analyse complexe aux surfaces d’aire minima, Topics in Complex Manifolds (Ed. H. Rossi), Les Presses de l’Universit ́e de Montr ́eal (1968), 59-81 (naid:10006413960)

• Michael Atiyah, Nigel Hitchin, Isadore Singer, Example 2 on p. 438 in: Self-Duality in Four-Dimensional Riemannian Geometry, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 362, No. 1711 (Sep. 12, 1978), pp. 425-461 (37 pages) (jstor:79638, doi:10.1098/rspa.1978.0143)

• Michael Atiyah, Section III.1 of: Geometry of Yang-Mills fields, Pisa, Italy: Sc. Norm. Sup. (1979) 98 (spire:150867, pdf)

• Robert Bryant, Section 1 of: Conformal and minimal immersions of compact surfaces into the 4-sphere, J. Differential Geom. Volume 17, Number 3 (1982), 455-473 (euclid:jdg/1214437137)

• Robert Bryant, Lie groups and twistor spaces, Duke Mathematical Journal 52 (1985), pp. 223–261, (euclid:dmj/1077304286)

• H. Blaine Lawson, Surfaces minimales et la construction de Calabi-Penrose, Séminaire Bourbaki : volume 1983/84, exposés 615-632, Astérisque no. 121-122 (1985), Talk no. 624, p. 197-211 (numdam:SB_1983-1984__26__197_0)

• Simon Salamon, p. 218-220 of: Harmonic and holomorphic maps, In: E. Vesentini (eds.) Geometry Seminar “Luigi Bianchi” II - 1984, Lecture Notes in Mathematics, vol 1164. Springer 1985 (doi:10.1007/BFb0081912)

• James Eells, Simon Salamon, Section 8 of: Twistorial construction of harmonic maps of surfaces into four-manifolds, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 4, Volume 12 (1985) no. 4, p. 589-640 (numdam:ASNSP_1985_4_12_4_589_0)

• Bonaventure Loo, The space of harmonic maps of $S^2$ into $S^4$, Transactions of the American Mathematical Society Vol. 313, No. 1 (1989) (jstor:2001066)

• José Seade, Alberto Verjovsky, Section 2 of: Higher dimensional complex Kleinian groups, Math Ann 322, 279–300 (2002) (doi:10.1007/s002080100247)

• Le, José Seade, Alberto Verjovsky, Section 4 of: Quadrics, orthogonal actions and involutions in complex projective space, L’Enseignement Mathématique, t. 49 (2003) (e-periodica:001:2003:49::488)

• Jonas Nordstrom, Calabi’s construction of Harmonic maps from $S^2$ to $S^n$, Lund University 2008 (pdf, pdf)

• Angel Cano, Juan Pablo Navarrete, José Seade, Section 10.1 in: Kleinian Groups and Twistor Theory, In: Complex Kleinian Groups, Progress in Mathematics, vol 303. Birkhäuser 2013 (doi:10.1007/978-3-0348-0481-3_10)

• John Armstrong, Simon Salamon, Twistor Topology of the Fermat Cubic, SIGMA 10 (2014), 061, 12 pages (arXiv:1310.7150)

• Bobby Acharya, Robert Bryant, Simon Salamon, A circle quotient of a $G_2$ cone, Differential Geometry and its Applications Volume 73, December 2020, 101681 (arXiv:1910.09518, doi:10.1016/j.difgeo.2020.101681)

In noncommutative geometry:

• Simon Brain, Giovanni Landi, Differential and Twistor Geometry of the Quantum Hopf Fibration, Commun. Math. Phys. 315 (2012):489-530 (arXiv:1103.0419)

In higher dimensions:

Over $\mathbb{H}P^3$:

• Bonaventure Loo, Alberto Verjovsky, On quotients of Hopf fibrations, in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 26 (1994), pp. 103-108 (hdl:10077/4637, pdf)