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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 22nd 2016
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   Author: Urs
   Format: MarkdownItexDennis Borisov kindly highlights this most remarkable article to me: * [[Naichung Conan Leung]], _Riemannian Geometry Over Different Normed Division Algebras_, J. Differential Geom. Volume 61, Number 2 (2002), 289-333. ([publisher page](http://projecteuclid.org/euclid.jdg/1090351387)) First of all it establishes this table here, which I gave an entry _[[normed division algebra Riemannian geometry -- table]]_: | [[normed division algebra]] | $\mathbb{A}$ | Riemannian $\mathbb{A}$-manifolds | Special Riemannian $\mathbb{A}$-manifolds | |--|--------------|-----------------------------------|------------------| | [[real numbers]] | $\mathbb{R}$ | [[Riemannian manifold]] | [[orientation|oriented]] [[Riemannian manifold]] | | [[complex numbers]] | $\mathbb{C}$ | [[Kähler manifold]] | [[Calabi-Yau manifold]] | | [[quaternions]] | $\mathbb{H}$ | [[quaternion-Kähler manifold]] | [[hyperkähler manifold]] | | [[octonions]] | $\mathbb{O}$ | [[Spin(7)-manifold]] | [[G2-manifold]] | But it goes beyond that to discuss the relevant connections etc. For the moment I have included the above table in some of the relevant entries.

   Dennis Borisov kindly highlights this most remarkable article to me:

   • Naichung Conan Leung, Riemannian Geometry Over Different Normed Division Algebras, J. Differential Geom. Volume 61, Number 2 (2002), 289-333. (publisher page)

   First of all it establishes this table here, which I gave an entry normed division algebra Riemannian geometry – table:

   normed division algebra	$\mathbb{A}$	Riemannian $\mathbb{A}$-manifolds	Special Riemannian $\mathbb{A}$-manifolds
   real numbers	$\mathbb{R}$	Riemannian manifold	oriented Riemannian manifold
   complex numbers	$\mathbb{C}$	Kähler manifold	Calabi-Yau manifold
   quaternions	$\mathbb{H}$	quaternion-Kähler manifold	hyperkähler manifold
   octonions	$\mathbb{O}$	Spin(7)-manifold	G2-manifold

   But it goes beyond that to discuss the relevant connections etc. For the moment I have included the above table in some of the relevant entries.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 22nd 2016
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   Author: David_Corfield
   Format: MarkdownItexIs there anything particularly important in the article you can point out to us to notice? Is it the common treatment across normed division algebras?

   Is there anything particularly important in the article you can point out to us to notice? Is it the common treatment across normed division algebras?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 23rd 2016
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   Author: Urs
   Format: MarkdownItexIt's that he also gets all the right instanton conditions on the exceptional connections, all unified. All the structure that ever appears in M/F-theory compactifications on G2/CY4-fibers, including all the moduli, seems to find a single unified home in this perspective. Boris and I are running a secret F-theory seminar on this and related matters. I have to go offline right now and will be all busy lecturing next week on something else. But then I'll come back to this story here and will put more details on the nLab.

   It’s that he also gets all the right instanton conditions on the exceptional connections, all unified. All the structure that ever appears in M/F-theory compactifications on G2/CY4-fibers, including all the moduli, seems to find a single unified home in this perspective.

   Boris and I are running a secret F-theory seminar on this and related matters. I have to go offline right now and will be all busy lecturing next week on something else. But then I’ll come back to this story here and will put more details on the nLab.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeDec 9th 2016
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   Author: David_Corfield
   Format: MarkdownItexIs there some relation between the normed division algebras giving rise to the “trunk” of the brane bouquet ([M-Theory from the Superpoint](https://ncatlab.org/schreiber/show/M-Theory+from+the+Superpoint)), and their appearance in this thread in terms of #3? Though I guess they can appear for different reasons, as in the [[magic pyramid]].

   Is there some relation between the normed division algebras giving rise to the “trunk” of the brane bouquet (M-Theory from the Superpoint), and their appearance in this thread in terms of #3? Though I guess they can appear for different reasons, as in the magic pyramid.