Adjoint Fields of Electromagnetic and Gravitational Fields in the Curved Spaces

J. C. Maxwell introduced the quaternion to describe the feature of electromagnetic field. This method illumines other scholars to bring the quaternion space into the gravitational theory. Currently the octonion space is able to depict the characteristics of electromagnetic field and gravitational field simultaneously. Further the octonion space can escalate the physical content of these two fields. The character of octonion operator allows each field in the octonion space to possess adjunctively the adjoint field. In the octonion space, the electromagnetic field has one adjoint field. The adjoint field of electromagnetic field possesses some features of gravitational field, and may be one candidate of the dark matter field. Similarly the gravitational field has one adjoint field, which provides some features of electromagnetic field. The octonion space can be chosen as the flat space or curved space. In the curved octonion space, the connection coefficient and the curvature have the influences on the physical quantities about the electromagnetic field, the gravitational field, and their adjoint fields.