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Generalized geometry (in Hitchin's sense, following Courant and Dorfman) is adapted to the physical motion of string-like particles in the same way that traditional geometry is adapted to the physical motion of point-like particles. More general generalized geometries are useful in connection with higher dimensional objects such as membranes (and hence also M-theory). Pavol Severa's first letter to Alan Weinstein is a nice early reference point for the basic idea.

Update 1. I won't be able to add significantly to this post until Monday, but I want to indicate some of the relationships between my answer (as it is and to come) and Urs'. Ignoring higher dimensional objects than strings for now, generalized geometry initially concerns geometry on the generalized tangent bundle $T\oplus T^*$ (where $T=TM$ is the tangent bundle of a manifold $M$). The bundle $T\oplus T^*$ has a natural symmetric form with respect to which both $T$ and $T^*$ are maximal isotropic.

However, generalized geometry takes the point of view that $T\oplus T^*$ is an extension of $T$ by $T^*$, and is thus an example of a Courant algebroid $CA$, in that there is a short exact sequence $0\to T^*\to CA\to T\to 0$, where $CA$ has a symmetric form and other structure (the Courant bracket) making it isomorphic to $T\oplus T^*$ for suitable isotropic splittings of the exact sequence. A Dirac structure is such an isotropic splitting.

From the naive $T\oplus T^*$ viewpoint, a Dirac structure is given by an orthogonal involution of $T\oplus T^*$ whose eigenspaces do not meet $T^*$. Generalized complex geometry is a subfield of generalized geometry, in which one studies orthogonal complex structures on $T\oplus T^*$ instead of involutions. However, there are interesting structures on $T\oplus T^*$ which involve neither Dirac structures nor generalized complex structures. This should not be surprising: there is more to ordinary geometry than involutions and complex structures.

Generalized geometry (in Hitchin's sense, following Courant and Dorfman) is adapted to the physical motion of string-like particles in the same way that traditional geometry is adapted to the physical motion of point-like particles. More general generalized geometries are useful in connection with higher dimensional objects such as membranes (and hence also M-theory). Pavol Severa's first letter to Alan Weinstein is a nice early reference point for the basic idea.

Update 1. I won't be able to add significantly to this post until Tuesday perhaps, but I want to indicate some of the relationships between my answer (as it is and to come) and Urs'. Ignoring higher dimensional objects than strings for now, generalized geometry initially concerns geometry on the generalized tangent bundle $T\oplus T^*$ (where $T=TM$ is the tangent bundle of a manifold $M$). The bundle $T\oplus T^*$ has a natural symmetric form with respect to which both $T$ and $T^*$ are maximal isotropic.

However, generalized geometry takes the point of view that $T\oplus T^*$ is an extension of $T$ by $T^*$, and is thus an example of a Courant algebroid $CA$, in that there is a short exact sequence $0\to T^*\to CA\to T\to 0$, where $CA$ has a symmetric form and other structure (the Courant bracket) making it isomorphic to $T\oplus T^*$ for suitable isotropic splittings of the exact sequence. A Dirac structure is such an isotropic splitting.

From the naive $T\oplus T^*$ viewpoint, a Dirac structure is given by an orthogonal involution of $T\oplus T^*$ whose eigenspaces do not meet $T^*$. Generalized complex geometry is a subfield of generalized geometry, in which one studies orthogonal complex structures on $T\oplus T^*$ instead of involutions. However, there are interesting structures on $T\oplus T^*$ which involve neither Dirac structures nor generalized complex structures. This should not be surprising: there is more to ordinary geometry than involutions and complex structures.

Generalized geometry (in Hitchin's sense, following Courant and Dorfman) is adapted to the physical motion of string-like particles in the same way that traditional geometry is adapted to the physical motion of point-like particles. More general generalized geometries are useful in connection with higher dimensional objects such as membranes (and hence also M-theory). Pavol Severa's first letter to Alan Weinstein is a nice early reference point for the basic idea.

Generalized geometry (in Hitchin's sense, following Courant and Dorfman) is adapted to the physical motion of string-like particles in the same way that traditional geometry is adapted to the physical motion of point-like particles. More general generalized geometries are useful in connection with higher dimensional objects such as membranes (and hence also M-theory). Pavol Severa's first letter to Alan Weinstein is a nice early reference point for the basic idea.

Update 1. I won't be able to add significantly to this post until Monday, but I want to indicate some of the relationships between my answer (as it is and to come) and Urs'. Ignoring higher dimensional objects than strings for now, generalized geometry initially concerns geometry on the generalized tangent bundle $T\oplus T^*$ (where $T=TM$ is the tangent bundle of a manifold $M$). The bundle $T\oplus T^*$ has a natural symmetric form with respect to which both $T$ and $T^*$ are maximal isotropic.

However, generalized geometry takes the point of view that $T\oplus T^*$ is an extension of $T$ by $T^*$, and is thus an example of a Courant algebroid $CA$, in that there is a short exact sequence $0\to T^*\to CA\to T\to 0$, where $CA$ has a symmetric form and other structure (the Courant bracket) making it isomorphic to $T\oplus T^*$ for suitable isotropic splittings of the exact sequence. A Dirac structure is such an isotropic splitting.

From the naive $T\oplus T^*$ viewpoint, a Dirac structure is given by an orthogonal involution of $T\oplus T^*$ whose eigenspaces do not meet $T^*$. Generalized complex geometry is a subfield of generalized geometry, in which one studies orthogonal complex structures on $T\oplus T^*$ instead of involutions. However, there are interesting structures on $T\oplus T^*$ which involve neither Dirac structures nor generalized complex structures. This should not be surprising: there is more to ordinary geometry than involutions and complex structures.

Generalized geometry (in Hitchin's sense, following Courant and Dorfman) is adapted to the physical motion of string-like particles in the same way that traditional geometry is adapted to the physical motion of point-like particles. More general generalized geometries are useful in connection with higher dimensional objects such as membranes (and hence also M-theory). Pavol Severa's first letter to Alan Weinstein (http://sophia.dtp.fmph.uniba.sk/~severa/letters) is a good first reference point.

Generalized geometry (in Hitchin's sense, following Courant and Dorfman) is adapted to the physical motion of string-like particles in the same way that traditional geometry is adapted to the physical motion of point-like particles. More general generalized geometries are useful in connection with higher dimensional objects such as membranes (and hence also M-theory). Pavol Severa's first letter to Alan Weinstein is a nice early reference point for the basic idea.