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We make the first attempt to define S-transformations for CFT$_2$ as linearmappings from closed to open sector linear spaces. The definition is based onclosed-open sector linear space isomorphisms and boundary conditioncompleteness. Diagonal RCFTs can be applied to our definitionstraight-forwardly, while more classes of CFT$_2$ are expected to beapplicable. An unconventional open sector sewing, not among open sector sewingintroduced by Lewellen, rises naturally and generalizes the definition. Ourgeometrical approach, partially inspired by string field theory, reveals therelationship between algebraic information in CFT$_2$ and curvature onsurfaces.

S-transformations for CFT$_2$ as linear mappings from closed to open sector linear spaces