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In information Geometry, there is a definition of dual connection, which is: two affine connections $\nabla$ and $\nabla^*$ are called dual if they satisfied $$X(g(Y,Z))=g(\nabla_XY,Z)+g(X,\nabla^*_XZ)$$$$X(g(Y,Z))=g(\nabla_XY,Z)+g(Y,\nabla^*_XZ)$$ It seems that this definition depends on the metric $g$. My question is could we find another way to define the dual connections above, without the metric involved?

In information Geometry, there is a definition of dual connection, which is: two affine connections $\nabla$ and $\nabla^*$ are called dual if they satisfied $$X(g(Y,Z))=g(\nabla_XY,Z)+g(X,\nabla^*_XZ)$$ It seems that this definition depends on the metric $g$. My question is could we find another way to define the dual connections above, without the metric involved?

In information Geometry, there is a definition of dual connection, which is: two affine connections $\nabla$ and $\nabla^*$ are called dual if they satisfied $$X(g(Y,Z))=g(\nabla_XY,Z)+g(Y,\nabla^*_XZ)$$ It seems that this definition depends on the metric $g$. My question is could we find another way to define the dual connections above, without the metric involved?

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user64142
user64142

Dual connections for Information Geometry

In information Geometry, there is a definition of dual connection, which is: two affine connections $\nabla$ and $\nabla^*$ are called dual if they satisfied $$X(g(Y,Z))=g(\nabla_XY,Z)+g(X,\nabla^*_XZ)$$ It seems that this definition depends on the metric $g$. My question is could we find another way to define the dual connections above, without the metric involved?