Invariant Metrics

The idea to construct metrics invariant for suitable maps goes back to Bernhard Riemann. Among many others ideas which came to determine the subsequent development of the geometric approach to mathematics and its application it was expressed in his famous lecture “Über die Hypothesen, welche der Geometrie zu Grunde liegen” (1854). In a more precise form it is formulated in the so-called Erlanger program of Felix Klein in the latters inaugural lecture “Vergleichende Betrachtungen über neuere geometrische Forschungen” (1872), where invariants of various transformation groups are exhibited.