Alpha Connection
A non-metric connection is not determined by a metric tensor ; instead, it is and restricted by the requirement that the parallel transport between points and must be a linear combination of the base vectors in . Here,
expresses the parallel transport of as linear combination of the base vectors in, i.e. the new minus the change. Note that it is not a tensor (does not transform as a tensor).
For such a metric, one can construct a dual connection to make
- ,
for parallel transport using and .
For the mentioned -families the affine connection is called the -connection and can also be expressed in more ways.
For :
- is a metric connection and with .
- ,
i.e. is dual to with respect to the Fisher metric.
- If this is called -affine. Its dual is then -affine.
, i.e. 0-affine, and hence, i.e. 1-affine.
Read more about this topic: Information Geometry, Introduction
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