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s00mb

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- Kahler Manifold question

Hi, I have just finished studying Riemannian Geometry and was moving on to trying to figure out what a Kahler manifold is. Using wikipedia's definition(probably a bad idea to start with) it says "Equivalently, there is a complex structure

*J*on the tangent space of*X*at each point (that is, a real linear map from*TX*to itself with*J^*2 = −1) such that*J*preserves the metric*g*(meaning that*g*(*Ju*,*Jv*) =*g*(*u*,*v*)) and*J*is preserved by parallel transport. " I am not sure what it means by complex structure. Is j a subset of the tangent space with that property or is J the map itself J : TX -> TX with J^2=-1 ? I guess it could kind of be both but I was looking if someone could clarify this for me. Thanks.