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In an arbitrary axisymmetric stationary spacetime, we determine theexpression for the tangential velocity of test objects following a circularstable geodesic motion in the equatorial plane, as function of the metriccoefficients. Next, we impose the condition, observed in large samples of disksgalaxies, that the magnitude of such tangential velocity be radii independent,obtaining a constraint equation among the metric coefficients, and thusarriving to an iff condition: The tangential velocity of test particles isradii independent iff the metric coefficients satisfied the mentionedconstraint equation. Furthermore, for the static case, the constraint equationcan be easily integrated, leaving the spacetime at the equatorial planeessentially with only one independent metric coefficient. With the geometrythus fixed, we compute the Einstein tensor and equate it to and arbitrarystress energy tensor, in order to determined the type of energy-matter whichcould produce such a geometry. Within an approximation, we deduce a constraintequation among the components of the stress energy tensor. We test in thatconstraint equation several well known types of matter, which have beenproposed as dark matter candidates and are able to point for possible rightones. Finally, we also present the spherically symmetric static case and applythe mentioned procedure to perfect fluid stress energy tensor, recovering theNewtonian result as well as the one obtained in the axisymmetric case. We alsopresent arguments on the need to use GR to study non dust types of matter.

Geometric Conditions on the Type of Matter Determining the Flat Behavior of the Rotational Curves in Galaxies