J-holomorphic curves in nearly Kähler $$\mathbb {CP}^3$$ CP 3 are related to minimal surfaces in $$S^4$$ S 4 as well as associative submanifolds in $$\Lambda ^2_-(S^4)$$ Λ - 2 ( S 4 ) . We introduce the class of transverse J-holomorphic curves and establish a Bonnet-type theorem for them. We classify flat tori in $$S^4$$ S 4 and construct moment-type maps from $$\mathbb {CP}^3$$ CP 3 to relate them to the theory of $$\mathrm {U}(1)$$ U ( 1 ) -invariant minimal surfaces on $$S^4$$ S 4 .

J-holomorphic curves in nearly Kähler CP 3 are related to minimal surfaces in S 4 as well as associative submanifolds in Λ 2 − (S 4 ). We introduce the class of transverse J-holomorphic curves and establish a Bonnet-type theorem for them. We classify flat tori in S 4 and construct moment-type maps from CP 3 to relate them to the theory of U(1)-invariant minimal surfaces on S 4 . IntroductionIn symplectic geometry, the theory of J-holomorphic curves is a fundamental topic, which has for example led to the construction of Gromov-Witten invariants. From a Riemannian point of view, it is desirable to have examples of minimal surfaces in Einstein manifolds. One class of such examples comes from complex geometry by considering holomorphic curves in Einstein-Kähler manifolds. Nearly Kähler six-manifolds are Einstein with positive Einstein constant and J-holomorphic curves provide a class of examples of minimal surfaces inside them. However, they are neither symplectic nor complex and apart from local statements not much is known about J-holomorphic curves in general almost complex manifolds. One feature that is shared with the symplectic case is that the nearly Kähler structure equations guarantee that the volume remains constant on connected components of the moduli space of J-holomorphic curves [29]. Nevertheless, a general, in-depth theory of J-holomorphic curves in nearly Kähler manifolds seems out of reach at the moment, which is why most work has been concerned with studying them in the homogeneous examples. Historically, M = S 6 has been the example studied the most, setoff by Bryant's construction of torsion-free J-holomorphic curves as integrals of a holomorphic differential system on Gr(2, R 7 ) [10]. More recently, questions about certain components of the moduli space of J-holomorphic curves have been tackled [15].J-holomorphic curves in the twistor spaces CP 3 and the flag manifold F are related to special submanifolds in two different geometric settings. Firstly, it is a general result that the cone of a Jholomorphic curve in a nearly Kähler manifold M gives an associative submanifold in the G 2 -cone C(M ). Remarkably, J-holomorphic curves in CP 3 and F also give rise to to complete associative submanifolds in the Bryant-Salamon spaces Λ 2 − (S 4 ) and Λ 2 − (CP 2 ) [21]. Secondly, via the Eells-Salamon correspondence [14], J-holomorphic curves in the twistor spaces are in correspondence with minimal surfaces in S 4 and CP 2 . Somewhat similar to torsion-free curves in S 6 , the space CP 3 admits a particular class of J-holomorphic curves called superminimal curves. They can also be constructed via integrals of a holomorphic differential system and admit a Weierstraß parametrisation [9]. In contrast, have been studied with methods of integrable systems [26] as minimal surfaces in S 4 . The aim of this article is to describe various results on non-superminimal surfaces in S 4 from the twistor perspective, building on F. Xu's work on J-holomorphic curves in nearly Kähler CP 3 [30]. Adapting the twistor pers...

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We approach the well-studied problem of supervised group invariant and equivariant machine learning from the point of view of geometric topology. We propose a novel approach using a pre-processing step, which involves projecting the input data into a geometric space which parametrises the orbits of the symmetry group. This new data can then be the input for an arbitrary machine learning model (neural network, random forest, support-vector machine etc).We give an algorithm to compute the geometric projection, which is efficient to implement, and we illustrate our approach on some example machine learning problems (including the wellstudied problem of predicting Hodge numbers of CICY matrices), in each case finding an improvement in accuracy versus others in the literature. The geometric topology viewpoint also allows us to give a unified description of so-called intrinsic approaches to group equivariant machine learning, which encompasses many other approaches in the literature.