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Mtheorist
Mtheorist
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I have found two different definitions for integrable quaternionic structure in the literature, and I need to know if they agree with one another.

One definition that I have found (from Differential Geometry of Lightlike Submanifolds - Duggal, Sahin) is that for an almost quaternion manifold, integrable quaternionic structure implies that there exists coordinates in each coordinate neighborbood whereby the three almost complex structures $J_1,J_2,J_3$ (that define the almost quaternionic structure) take the form

$$ J_1= \left[\begin{array}{cccc} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1\\ 0 & 0 & 1 & 0 \end{array}\right] $$ $$ J_2= \left[\begin{array}{cccc} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \end{array}\right] $$

$$ J_3= \left[\begin{array}{cccc} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 \end{array}\right]. $$$$ J_3= \left[\begin{array}{cccc} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 \end{array}\right], $$ where $1$ is the identity matrix.

On the other hand, from this article, I have found a definition where integrable quaternionic structure implies that the Nijenhuis tensor for each of the almost complex structures given above vanish.

Do these definitions agree?

I have found two different definitions for integrable quaternionic structure in the literature, and I need to know if they agree with one another.

One definition that I have found (from Differential Geometry of Lightlike Submanifolds - Duggal, Sahin) is that for an almost quaternion manifold, integrable quaternionic structure implies that there exists coordinates in each coordinate neighborbood whereby the three almost complex structures $J_1,J_2,J_3$ (that define the almost quaternionic structure) take the form

$$ J_1= \left[\begin{array}{cccc} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1\\ 0 & 0 & 1 & 0 \end{array}\right] $$ $$ J_2= \left[\begin{array}{cccc} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \end{array}\right] $$

$$ J_3= \left[\begin{array}{cccc} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 \end{array}\right]. $$

On the other hand, from this article, I have found a definition where integrable quaternionic structure implies that the Nijenhuis tensor for each of the almost complex structures given above vanish.

Do these definitions agree?

I have found two different definitions for integrable quaternionic structure in the literature, and I need to know if they agree with one another.

One definition that I have found (from Differential Geometry of Lightlike Submanifolds - Duggal, Sahin) is that for an almost quaternion manifold, integrable quaternionic structure implies that there exists coordinates in each coordinate neighborbood whereby the three almost complex structures $J_1,J_2,J_3$ (that define the almost quaternionic structure) take the form

$$ J_1= \left[\begin{array}{cccc} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1\\ 0 & 0 & 1 & 0 \end{array}\right] $$ $$ J_2= \left[\begin{array}{cccc} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \end{array}\right] $$

$$ J_3= \left[\begin{array}{cccc} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 \end{array}\right], $$ where $1$ is the identity matrix.

On the other hand, from this article, I have found a definition where integrable quaternionic structure implies that the Nijenhuis tensor for each of the almost complex structures given above vanish.

Do these definitions agree?

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Mtheorist
Mtheorist
  • 1.2k
  • 6
  • 10

Do these definitions of integrable quaternionic structure agree?

I have found two different definitions for integrable quaternionic structure in the literature, and I need to know if they agree with one another.

One definition that I have found (from Differential Geometry of Lightlike Submanifolds - Duggal, Sahin) is that for an almost quaternion manifold, integrable quaternionic structure implies that there exists coordinates in each coordinate neighborbood whereby the three almost complex structures $J_1,J_2,J_3$ (that define the almost quaternionic structure) take the form

$$ J_1= \left[\begin{array}{cccc} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1\\ 0 & 0 & 1 & 0 \end{array}\right] $$ $$ J_2= \left[\begin{array}{cccc} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \end{array}\right] $$

$$ J_3= \left[\begin{array}{cccc} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 \end{array}\right]. $$

On the other hand, from this article, I have found a definition where integrable quaternionic structure implies that the Nijenhuis tensor for each of the almost complex structures given above vanish.

Do these definitions agree?