My talk will begin with a survey on the existence theorem of a neutral metric on a 4-manifold. The existence of a neutral metric on a 4-manifold is equivalent to the existence of a field 2-planes, and also to that of a pair of two kinds of almost complex structures. These facts are proved on the basis of a theorem of Hirzebruch and Hopf (1958), and of Donaldson’s classification of definite intersection forms (1983). I then consider the Goldberg Conjecture (1969) of indefinite metric version, and I report some counterexamples to the Conjecture of indefinite metric version. Also, future problems will be discussed.