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The answer is yes!(at least if quaternionic holomorphic geometry counts) "Qauternionic holomorphic geometry" provides a very elegant description of surfaces in 3- and 4-dimensional space.

The first paper in this field is more or less

Franz Pedit and Ulrich Pinkall, Quaternionic Analysis on Riemann Surfaces and Differential Geometry, in: Proocedings of the international con- gress of mathematicians, Berlin 1998, II Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, Extra Volume ICM 1998, 389-400. Click me

A good introduction to "Quaternionic holomorphic geometry" is given by

Francis E. Burstall, Dirk Ferus, Katrin Leschke, Franz Pedit and Ulrich Pinkall, Conformal geometry of surfaces in S4 and quaternions, Lecture Notes in Mathematics 1772, Springer-Verlag, Berlin, 2002. http://arxiv.org/abs/math/0002075

and

Dirk Ferus, Katrin Leschke, Franz Pedit and Ulrich Pinkall, Quaternionic Holomorphic Geometry: Plucker Formula, Dirac Eigenvalue Esitmates and Energy Estimates of Harmonic 2-Tori, Inventiones Mathematicae 146 (2001),no. 3, 507-593 arxiv.org/abs/math/0012238v1

But just to clarify things: "Of course, if such a theory has some sense, it cannot be the "obvious analog" of complex algebraic or analytic geometry, as theorems 1. and 2. above show."

This isn't really true, because the quaternionic holomorphic geometry developed in the papers above, is a kind of generalization of complex geometry, i.e. if you look how a "quaternionic holomorphic structure" is defined, you see that it is a kind of $\overline{\partial}$-Operator with some extra data (a so called Hopf field).

If you need more references, there are plenty available (just type "quaternionic holomorphic geometry" into google)

The answer is yes!(at least if quaternionic holomorphic geometry counts) "Quaternionic holomorphic geometry" provides a very elegant description of surfaces in 3- and 4-dimensional space.

The first paper in this field is more or less

Franz Pedit and Ulrich Pinkall, Quaternionic Analysis on Riemann Surfaces and Differential Geometry, in: Proocedings of the international con- gress of mathematicians, Berlin 1998, II Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, Extra Volume ICM 1998, 389-400. Click me

A good introduction to "Quaternionic holomorphic geometry" is given by

Francis E. Burstall, Dirk Ferus, Katrin Leschke, Franz Pedit and Ulrich Pinkall, Conformal geometry of surfaces in S4 and quaternions, Lecture Notes in Mathematics 1772, Springer-Verlag, Berlin, 2002. http://arxiv.org/abs/math/0002075

and

Dirk Ferus, Katrin Leschke, Franz Pedit and Ulrich Pinkall, Quaternionic Holomorphic Geometry: Plucker Formula, Dirac Eigenvalue Esitmates and Energy Estimates of Harmonic 2-Tori, Inventiones Mathematicae 146 (2001),no. 3, 507-593 arxiv.org/abs/math/0012238v1

But just to clarify things: "Of course, if such a theory has some sense, it cannot be the "obvious analog" of complex algebraic or analytic geometry, as theorems 1. and 2. above show."

This isn't really true, because the quaternionic holomorphic geometry developed in the papers above, is a kind of generalization of complex geometry, i.e. if you look how a "quaternionic holomorphic structure" is defined, you see that it is a kind of $\overline{\partial}$-Operator with some extra data (a so called Hopf field).

If you need more references, there are plenty available (just type "quaternionic holomorphic geometry" into google)

The answer is yes!(at least if quaternionic holomorphic geomerty counts) "Qauternionic holomorphic geometry" provides a very elegant description of surfaces in 3- and 4-dimensional space.

The first paper is more or less

Franz Pedit and Ulrich Pinkall, Quaternionic Analysis on Riemann Surfaces and Differential Geometry, in: Proocedings of the international con- gress of mathematicians, Berlin 1998, II Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, Extra Volume ICM 1998, 389-400. Click me

A good introduction to "quaternionic holomorphic geometry" is given by

Francis E. Burstall, Dirk Ferus, Katrin Leschke, Franz Pedit and Ulrich Pinkall, Conformal geometry of surfaces in S4 and quaternions, Lecture Notes in Mathematics 1772, Springer-Verlag, Berlin, 2002. http://arxiv.org/abs/math/0002075

and

Dirk Ferus, Katrin Leschke, Franz Pedit and Ulrich Pinkall, Quater- nionic Holomorphic Geometry: Plucker Formula, Dirac Eigenvalue Esitmates and Energy Estimates of Harmonic 2-Tori, Inventiones Mathematicae 146 (2001),no. 3, 507-593 arxiv.org/abs/math/0012238v1

But just to clarify things: "Of course, if such a theory has some sense, it cannot be the "obvious analog" of complex algebraic or analytic geometry, as theorems 1. and 2. above show."

This isn't really true, because the quaternionic holomorphic geometry developed in the papers above, is a kind of generalization of complex geometry, i.e. if you look how "quaternionic holomorphic structure" is defined, you see that is a kind of $\overline{\partial}$-Operator with some extra data (a so called Hopf field).

If you need more references, there are plenty available (just type "quaternionic holomorphic geometry" into google)

The answer is yes!(at least if quaternionic holomorphic geometry counts) "Qauternionic holomorphic geometry" provides a very elegant description of surfaces in 3- and 4-dimensional space.

The first paper in this field is more or less

Franz Pedit and Ulrich Pinkall, Quaternionic Analysis on Riemann Surfaces and Differential Geometry, in: Proocedings of the international con- gress of mathematicians, Berlin 1998, II Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, Extra Volume ICM 1998, 389-400. Click me

A good introduction to "Quaternionic holomorphic geometry" is given by

Francis E. Burstall, Dirk Ferus, Katrin Leschke, Franz Pedit and Ulrich Pinkall, Conformal geometry of surfaces in S4 and quaternions, Lecture Notes in Mathematics 1772, Springer-Verlag, Berlin, 2002. http://arxiv.org/abs/math/0002075

and

Dirk Ferus, Katrin Leschke, Franz Pedit and Ulrich Pinkall, Quaternionic Holomorphic Geometry: Plucker Formula, Dirac Eigenvalue Esitmates and Energy Estimates of Harmonic 2-Tori, Inventiones Mathematicae 146 (2001),no. 3, 507-593 arxiv.org/abs/math/0012238v1

But just to clarify things: "Of course, if such a theory has some sense, it cannot be the "obvious analog" of complex algebraic or analytic geometry, as theorems 1. and 2. above show."

This isn't really true, because the quaternionic holomorphic geometry developed in the papers above, is a kind of generalization of complex geometry, i.e. if you look how a "quaternionic holomorphic structure" is defined, you see that it is a kind of $\overline{\partial}$-Operator with some extra data (a so called Hopf field).

If you need more references, there are plenty available (just type "quaternionic holomorphic geometry" into google)

The answer is yes!(at least if quaternionic holomorphic geomerty counts) "Qauternionic holomorphic geometry" provides a very elegant description of surfaces in 3- and 4-dimensional space.

The first paper is more or less

Franz Pedit and Ulrich Pinkall, Quaternionic Analysis on Riemann Surfaces and Differential Geometry, in: Proocedings of the international con- gress of mathematicians, Berlin 1998, II Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, Extra Volume ICM 1998, 389-400. [Click me][1]

A good introduction to "quaternionic holomorphic geometry" is given by

Francis E. Burstall, Dirk Ferus, Katrin Leschke, Franz Pedit and Ulrich Pinkall, Conformal geometry of surfaces in S4 and quaternions, Lecture Notes in Mathematics 1772, Springer-Verlag, Berlin, 2002.

and

Dirk Ferus, Katrin Leschke, Franz Pedit and Ulrich Pinkall, Quater- nionic Holomorphic Geometry: Plucker Formula, Dirac Eigenvalue Esitmates and Energy Estimates of Harmonic 2-Tori, Inventiones Mathematicae 146 (2001),no. 3, 507-593 [arxiv.org/abs/math/0012238v1][2]

But just to clarify things: "Of course, if such a theory has some sense, it cannot be the "obvious analog" of complex algebraic or analytic geometry, as theorems 1. and 2. above show."

This isn't true, because the quaternionic homolomorphic geometry developed in the papers above, is a kind of generalisation of complex geometry, i.e. if you look how "quaternionic holomorphic structure" is defined, you see that is a kind of $\overline{\partial}$-Operator with some extra data.

If you need more references, there are plenty avilable [1]: http://www.math.tu-berlin.de/~pinkall/forDownload/Pinkall.MAN.pdf [2]: http://arxiv.org/abs/math/0012238v1

The answer is yes!(at least if quaternionic holomorphic geomerty counts) "Qauternionic holomorphic geometry" provides a very elegant description of surfaces in 3- and 4-dimensional space.

The first paper is more or less

Franz Pedit and Ulrich Pinkall, Quaternionic Analysis on Riemann Surfaces and Differential Geometry, in: Proocedings of the international con- gress of mathematicians, Berlin 1998, II Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, Extra Volume ICM 1998, 389-400. Click me

A good introduction to "quaternionic holomorphic geometry" is given by

Francis E. Burstall, Dirk Ferus, Katrin Leschke, Franz Pedit and Ulrich Pinkall, Conformal geometry of surfaces in S4 and quaternions, Lecture Notes in Mathematics 1772, Springer-Verlag, Berlin, 2002. http://arxiv.org/abs/math/0002075

and

Dirk Ferus, Katrin Leschke, Franz Pedit and Ulrich Pinkall, Quater- nionic Holomorphic Geometry: Plucker Formula, Dirac Eigenvalue Esitmates and Energy Estimates of Harmonic 2-Tori, Inventiones Mathematicae 146 (2001),no. 3, 507-593 arxiv.org/abs/math/0012238v1

But just to clarify things: "Of course, if such a theory has some sense, it cannot be the "obvious analog" of complex algebraic or analytic geometry, as theorems 1. and 2. above show."

This isn't really true, because the quaternionic holomorphic geometry developed in the papers above, is a kind of generalization of complex geometry, i.e. if you look how "quaternionic holomorphic structure" is defined, you see that is a kind of $\overline{\partial}$-Operator with some extra data (a so called Hopf field).

If you need more references, there are plenty available (just type "quaternionic holomorphic geometry" into google)