  Based on the concept of curved spacetime in Einsteinian General Relativity,
the field theories and their quantum theories in the curved octonion spaces etc
are discussed. The research results discover the close relationships of the
curved spacetimes with the field theories and quantum theories. In the field
theories of curved spacetimes, the curvatures have direct effect on field
strength and field source etc. While the curvatures have direct impact on wave
functions and quantum equations etc in the quantum theories of curved
spacetimes. The research results discover that some abnormal phenomena of field
source particles could be explained by the field theories or quantum theories
in the curved spacetimes.
