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1. • CommentRowNumber1.
   • CommentAuthorJames Francese
   • CommentTimeMar 20th 2019
   • PermaLink
   Author: James Francese
   Format: MarkdownItexI've created a page for quaternionic manifolds, linked from quaternion-Kähler page. Basic references and discussion of main definition re Cauchy-Feuter calculus. Comparison to hypercomplex structure. <a href="https://ncatlab.org/nlab/revision/quaternionic+manifold/1">v1</a>, <a href="https://ncatlab.org/nlab/show/quaternionic+manifold">current</a>

   I’ve created a page for quaternionic manifolds, linked from quaternion-Kähler page. Basic references and discussion of main definition re Cauchy-Feuter calculus. Comparison to hypercomplex structure.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 21st 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks! And I have added now the link going the other way, too. Might you have energy to add a sentence putting the two in relation?

   Thanks! And I have added now the link going the other way, too.

   Might you have energy to add a sentence putting the two in relation?

3. • CommentRowNumber3.
   • CommentAuthorJames Francese
   • CommentTimeMar 21st 2019
   • PermaLink
   Author: James Francese
   Format: MarkdownItexGreat idea, I've added a comment connecting their notations and a statement of why quaternionic-Kähler manifolds are indeed quaternionic. <a href="https://ncatlab.org/nlab/revision/diff/quaternionic+manifold/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/quaternionic+manifold/4">v4</a>, <a href="https://ncatlab.org/nlab/show/quaternionic+manifold">current</a>

   Great idea, I’ve added a comment connecting their notations and a statement of why quaternionic-Kähler manifolds are indeed quaternionic.

   diff, v4, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 21st 2019
   • (edited Mar 21st 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks! I gave the discussion of "qK is q" a numbered Example-environment ([here](https://ncatlab.org/nlab/show/quaternionic+manifold#QuaternionKaehlermanifoldsAreQuaternionicManifolds)) and placed that inside an Examples section. Also, I created a Properties-section and made your two stub sections "Holonomy" and "Twistor space" be subsections of that. <a href="https://ncatlab.org/nlab/revision/diff/quaternionic+manifold/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/quaternionic+manifold/5">v5</a>, <a href="https://ncatlab.org/nlab/show/quaternionic+manifold">current</a>

   Thanks!

   I gave the discussion of “qK is q” a numbered Example-environment (here) and placed that inside an Examples section.

   Also, I created a Properties-section and made your two stub sections “Holonomy” and “Twistor space” be subsections of that.

   diff, v5, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMar 21st 2019
   • (edited Mar 21st 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexBy the way, I would tend to make your "Discussion"-subsection be a numbered Remark inside the "Definition"-section instead. (I find it helpful to think in terms of textbook organization: There we would rarely make a comment on a definition be its own subsection.)

   By the way, I would tend to make your “Discussion”-subsection be a numbered Remark inside the “Definition”-section instead.

   (I find it helpful to think in terms of textbook organization: There we would rarely make a comment on a definition be its own subsection.)

6. • CommentRowNumber6.
   • CommentAuthorJames Francese
   • CommentTimeMar 21st 2019
   • PermaLink
   Author: James Francese
   Format: MarkdownItexThanks Urs. And agreed, that is much more natural. I'm paying attention to the changes you've been making as I add material; as I learn more itex + markdown and nLab style I can follow suit.

   Thanks Urs. And agreed, that is much more natural. I’m paying attention to the changes you’ve been making as I add material; as I learn more itex + markdown and nLab style I can follow suit.

7. • CommentRowNumber7.
   • CommentAuthorJames Francese
   • CommentTimeMar 21st 2019
   • PermaLink
   Author: James Francese
   Format: MarkdownItexAdded more classical and a few specialty references, including Bonan's seminal paper. <a href="https://ncatlab.org/nlab/revision/diff/quaternionic+manifold/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/quaternionic+manifold/6">v6</a>, <a href="https://ncatlab.org/nlab/show/quaternionic+manifold">current</a>

   Added more classical and a few specialty references, including Bonan’s seminal paper.

   diff, v6, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJul 15th 2020
   • (edited Jul 15th 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexcompleted the pointer to * [[Dmitri V. Alekseevsky]], [[Stefano Marchiafava]], _Quaternionic-like structures on a manifold: Note I. 1-integrability and integrability conditions_, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993) Volume: 4, Issue: 1, page 43-52 ([dml:244082](https://eudml.org/doc/244082)) and added pointer to * [[Dmitri V. Alekseevsky]], [[Stefano Marchiafava]], _Quaternionic-like structures on a manifold: Note II. Automorphism groups and their interrelations_, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993) Volume: 4, Issue: 1, page 53-61 ([dml:244299](https://eudml.org/doc/244299)) <a href="https://ncatlab.org/nlab/revision/diff/quaternionic+manifold/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/quaternionic+manifold/8">v8</a>, <a href="https://ncatlab.org/nlab/show/quaternionic+manifold">current</a>

   completed the pointer to

   • Dmitri V. Alekseevsky, Stefano Marchiafava, Quaternionic-like structures on a manifold: Note I. 1-integrability and integrability conditions, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993) Volume: 4, Issue: 1, page 43-52 (dml:244082)

   and added pointer to

   • Dmitri V. Alekseevsky, Stefano Marchiafava, Quaternionic-like structures on a manifold: Note II. Automorphism groups and their interrelations, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993) Volume: 4, Issue: 1, page 53-61 (dml:244299)

   diff, v8, current

9. • CommentRowNumber9.
   • CommentAuthoraleks_kleyn
   • CommentTimeNov 6th 2024
   • (edited Nov 6th 2024)
   • PermaLink
   Author: aleks_kleyn
   Format: MarkdownItexBecause you consider quaternionic manifold, then tangent plane is vector space over quaternion algebra. In vector space over quaternion algebra, we need to distinguish linear map and homomorphism. When I select basis in vector space, automorphism is still presented by matrix with quaternion entries. However, to represent linear map of vector space over quaternion algebra, I need matrix with entries from algebra $H\otimes H$. Now we can see the structure of connection $\Gamma^i_{kj}$ where index $k$ is responsible for displacement along manifold and indices $i$, $j$ are responsible for transformation of vector in tangent plane. I will start from transformation of vector. When I move from one point to another, transformation of vector is due to change of basis in tangent plane. Therefore indices $i$, $j$ are responsible for homomorphism. The story about index $k$ is different. I can consider set of bases in different tangent planes as almost everywhere continues map, displacement along manifold as differential and connection as derivative. Therefore, connection is responsible for linear transformation. From this it follows that connection is set of tensors like $\Gamma_{s0}^{}{}^i_{kj}\otimes\Gamma_{s1}^{}{}^i_{kj}$ (we have here sum over index s) and corresponding change of vector has form $$dv^i=\Gamma_{s0}^{}{}^i_{kj}dx^k\Gamma_{s1}^{}{}^i_{kj}v^j$$

   Because you consider quaternionic manifold, then tangent plane is vector space over quaternion algebra. In vector space over quaternion algebra, we need to distinguish linear map and homomorphism. When I select basis in vector space, automorphism is still presented by matrix with quaternion entries. However, to represent linear map of vector space over quaternion algebra, I need matrix with entries from algebra $H\otimes H$. Now we can see the structure of connection $\Gamma^i_{kj}$ where index $k$ is responsible for displacement along manifold and indices $i$, $j$ are responsible for transformation of vector in tangent plane. I will start from transformation of vector. When I move from one point to another, transformation of vector is due to change of basis in tangent plane. Therefore indices $i$, $j$ are responsible for homomorphism. The story about index $k$ is different. I can consider set of bases in different tangent planes as almost everywhere continues map, displacement along manifold as differential and connection as derivative. Therefore, connection is responsible for linear transformation. From this it follows that connection is set of tensors like $\Gamma_{s0}^{}{}^i_{kj}\otimes\Gamma_{s1}^{}{}^i_{kj}$ (we have here sum over index s) and corresponding change of vector has form

   $$dv^i=\Gamma_{s0}^{}{}^i_{kj}dx^k\Gamma_{s1}^{}{}^i_{kj}v^j$$