Remember the first time you did geometry. I bet it was two-dimensional geometry. You worked on the real plane and you measured distances, angles, areas... The real plane was the simplest place to do that. A one-dimensional line was rather abstract and the three-dimensional space felt way too big at that moment.
Later in life you came across surfaces. If you wanted to do something similar to what you did on the real plane, say, measure an angle between two curves on the surface, how did you do it? Well, at the intersection of these two curves you put a plane, the tangent plane, and each curve defined a vector on that plane (the tangent vector of the curve at that point). Then you could measure the angle between those two vectors as you did on the real plane!
Actually, what I have described is the way most of differential geometry has been created. The tangent plane of a surface or the tangent space of a manifold \(M\) have been used to define metrics, complex structures, symplectic structures... How many times have you seen \( TM \) around?
Generalized geometry proposes replacing \( TM \) by \( TM \oplus T^*M \) and redoing geometry from the beginning. Let us show what this means with an example. In order to make it shorter, let me use \( T \) for \( TM \), \( T^* \) for \( T^*M \) and use \( + \) instead of \( \oplus \) from now on.
Ok, first generalization. An almost complex structure is defined as a bundle map \( J: T \to T \) such that \( J^2=-\textrm{Id} \), so we will define a generalized almost complex structure as a map $$ \mathcal{J}: T + T^* \to T + T^* $$ such that \( \mathcal{J}^2=-\textrm{Id} \), and which satisfies another property that we omit for now.
One can see that almost complex structures fit in our new generalized setting. Given \( J \), we produce $$ \mathcal{J} = \left( \begin{array}{cc} -J & 0 \\ 0 & J^* \end{array} \right) $$ by using the diagonal entries.
What if we use the off-diagonal entries? We need a map \( T\to T^* \) and a map \( T^* \to T \). What about a pre-symplectic structure \( \omega \) and its inverse?
$$ \mathcal{J}_\omega = \left( \begin{array}{cc} 0 & -\omega^{-1} \\ \omega & 0 \end{array} \right).$$ Thus, at least intuitively for now, both complex and symplectic structures are particular cases of generalized complex structures. Generalized geometry provides, in first place, an unifying way of looking at previous structures.
But not only that. Generalized geometry also introduces genuinely new structures: there are compact manifolds which are neither complex nor symplectic, but still admit a generalized complex structure, for instance \( 3\mathbb{C} P^2 \# 19 \overline{\mathbb{C} P^2} \).
A third remarkable phenomenon in generalized geometry is the revival of previously known but somehow forgotten structures. This is the case of generalized Kahler structures. Generalized Kahler geometry was defined and shown to be equivalent to a bihermitian geometry defined in 1984. The statement of this equivalence was indeed followed by many publications about this subject.
Let us put some names to this. Generalized Geometry, as we are presenting it, was introduced by Nigel Hitchin around 2002. He dealt with Generalized Calabi-Yau manifolds and his PhD student Marco Gualtieri developed Generalized Complex and Kahler Geometry. Then, many other people and more of Hitchin's students, like Gil Cavalcanti, kept working on this. Actually, Cavalcanti found, together with Gualtieri, the example \( 3\mathbb{C} P^2 \# 19 \overline{\mathbb{C} P^2} \). However, as we will see, many of the objects we will be dealing with were already there. Generalized complex structures are, in particular, Dirac structures, a concept introduced by Ted Courant and Alan Weinstein around 1986, which unified Poisson structures and closed \(2\)-forms. We will also talk about the Dorfmann product, introduced by Irene Dorfman herself in 1987, and about Courant algebroids, a structure introduced by Weinstein together with Zhang-Ju Liu and Ping Xu in 1997. As you can see, all of these were ready before the 21st century!
Later in life you came across surfaces. If you wanted to do something similar to what you did on the real plane, say, measure an angle between two curves on the surface, how did you do it? Well, at the intersection of these two curves you put a plane, the tangent plane, and each curve defined a vector on that plane (the tangent vector of the curve at that point). Then you could measure the angle between those two vectors as you did on the real plane!
Actually, what I have described is the way most of differential geometry has been created. The tangent plane of a surface or the tangent space of a manifold \(M\) have been used to define metrics, complex structures, symplectic structures... How many times have you seen \( TM \) around?
Generalized geometry proposes replacing \( TM \) by \( TM \oplus T^*M \) and redoing geometry from the beginning. Let us show what this means with an example. In order to make it shorter, let me use \( T \) for \( TM \), \( T^* \) for \( T^*M \) and use \( + \) instead of \( \oplus \) from now on.
Ok, first generalization. An almost complex structure is defined as a bundle map \( J: T \to T \) such that \( J^2=-\textrm{Id} \), so we will define a generalized almost complex structure as a map $$ \mathcal{J}: T + T^* \to T + T^* $$ such that \( \mathcal{J}^2=-\textrm{Id} \), and which satisfies another property that we omit for now.
One can see that almost complex structures fit in our new generalized setting. Given \( J \), we produce $$ \mathcal{J} = \left( \begin{array}{cc} -J & 0 \\ 0 & J^* \end{array} \right) $$ by using the diagonal entries.
What if we use the off-diagonal entries? We need a map \( T\to T^* \) and a map \( T^* \to T \). What about a pre-symplectic structure \( \omega \) and its inverse?
$$ \mathcal{J}_\omega = \left( \begin{array}{cc} 0 & -\omega^{-1} \\ \omega & 0 \end{array} \right).$$ Thus, at least intuitively for now, both complex and symplectic structures are particular cases of generalized complex structures. Generalized geometry provides, in first place, an unifying way of looking at previous structures.
But not only that. Generalized geometry also introduces genuinely new structures: there are compact manifolds which are neither complex nor symplectic, but still admit a generalized complex structure, for instance \( 3\mathbb{C} P^2 \# 19 \overline{\mathbb{C} P^2} \).
A third remarkable phenomenon in generalized geometry is the revival of previously known but somehow forgotten structures. This is the case of generalized Kahler structures. Generalized Kahler geometry was defined and shown to be equivalent to a bihermitian geometry defined in 1984. The statement of this equivalence was indeed followed by many publications about this subject.
Let us put some names to this. Generalized Geometry, as we are presenting it, was introduced by Nigel Hitchin around 2002. He dealt with Generalized Calabi-Yau manifolds and his PhD student Marco Gualtieri developed Generalized Complex and Kahler Geometry. Then, many other people and more of Hitchin's students, like Gil Cavalcanti, kept working on this. Actually, Cavalcanti found, together with Gualtieri, the example \( 3\mathbb{C} P^2 \# 19 \overline{\mathbb{C} P^2} \). However, as we will see, many of the objects we will be dealing with were already there. Generalized complex structures are, in particular, Dirac structures, a concept introduced by Ted Courant and Alan Weinstein around 1986, which unified Poisson structures and closed \(2\)-forms. We will also talk about the Dorfmann product, introduced by Irene Dorfman herself in 1987, and about Courant algebroids, a structure introduced by Weinstein together with Zhang-Ju Liu and Ping Xu in 1997. As you can see, all of these were ready before the 21st century!
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