Abstract
Considering the possible types of numbers we have \(\mathbb {R}, \mathbb {C}, \mathbb {H}\), or \(\mathbb {O}\). This is a message for geometry. Keeping the fundamental idea that a geometrical space should be viewed as a manifold, constructed by means of an atlas of open charts, the local coordinates could be chosen not only as real numbers but also as complex, quaternionic or even octonionic numbers. Yet an important lesson is immediately learnt from the story told in my other book (Fré, A conceptual history of symmetry from Plato to the Superworld, Springer, Berlin, 2018, [1]), twin of the present one: the possible numbers are, anyhow, division algebras over the reals, whose classification is due to Frobenius, so that the real structure remains the basis for everything.
Mathematics, however, is, as it were, its own explanation; this, although it may seem hard to accept, is nevertheless true, for the recognition that a fact is so is the cause upon which we base the proof.
Girolamo Cardano
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- 1.
Complex analytic manifold means a manifold whose transition functions in the intersection of two charts are holomorphic functions of the local coordinates.
- 2.
- 3.
The development of de Rham cohomology and of characteristic classes is historically reviewed in the twin book to this one [1], within the general frame of the evolution of geometry in the XXth century.
- 4.
The interesting history of the Chern–Weil homomorphism, independently discovered by the two great mathematicians in the years of World War II, is reported in the twin book [1].
- 5.
We stress the word “formal eigenvalues” because the correct framework to understand these eigenvalues is the “splitting principle”, which, for convenience, is mentioned after the Eq. (2.7.59).
- 6.
For Kähler’s life, his relations with Chern and other outstanding mathematicians and for the conceptual development of Kähler metrics we refer the reader to the twin book [1].
- 7.
Special Kähler geometry will be discussed in Chap. 4, yet we anticipate here that it is the geometrical structure imposed by \(\mathscr {N}=2\) supersymmetry on the scalars belonging to vector multiplets (the scalar partners of the gauge vectors). In our notations the Special Kähler manifold which describes the interaction of vector multiplets is denoted \(\widehat{\mathscr {SK} }\) and all the Special Geometry Structures are endowed with a hat in order to distinguish this Special Kähler manifold from the other one which is incapsulated into the Quaternionic Kähler manifold \(\mathscr {QM}\) describing the hypermultiplets when this latter happens to be in the image of the c-map. For all these concepts we refer the reader to Chap. 4. They are not necessary to understand the present constructions, yet they were essential part for their establishment in the original papers mentioned here above.
- 8.
- 9.
- 10.
Let us stress that this is true for Hadamard manifolds and possibly for \(\mathrm {CAT}(k)\) manifolds, in any case for simple connected manifolds. In the presence of a non trivial fundamental group the presence of a fixed point is not necessary in order to establish the compact nature of the isometry group.
- 11.
Note that \(\left[ -\infty ,0\right] \) as range of the C-coordinate is conventional. Were it to be \(\left[ \infty ,0\right] \), we could just replace \(C \,\rightarrow \, -\,C\) which is always possible since the Kähler metric is given by Eq. 3.8.16.
- 12.
The factor 2 introduced in this equation is chosen in order to have a normalization of what we name curvature that agrees with the normalization used in several papers of the physical literature.
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Fré, P.G. (2018). Complex and Quaternionic Geometry. In: Advances in Geometry and Lie Algebras from Supergravity. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-74491-9_3
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