  The paper aims to extend major equations in the electromagnetic and
gravitational theories from the flat space into the complex octonion curved
space. Maxwell applied simultaneously the quaternion analysis and vector
terminology to describe the electromagnetic theory. It inspires subsequent
scholars to study the electromagnetic and gravitational theories with the
complex quaternions/octonions. Furthermore Einstein was the first to depict the
gravitational theory by means of tensor analysis and curved four-space-time.
Nowadays some scholars investigate the electromagnetic and gravitational
properties making use of the complex quaternion/octonion curved space. From the
orthogonality of two complex quaternions, it is possible to define the
covariant derivative of the complex quaternion curved space, describing the
gravitational properties in the complex quaternion curved space. Further it is
possible to define the covariant derivative of the complex octonion curved
space by means of the orthogonality of two complex octonions, depicting
simultaneously the electromagnetic and gravitational properties in the complex
octonion curved space. The result reveals that the connection coefficient and
curvature of the complex octonion curved space will exert an influence on the
field strength and field source of the electromagnetic and gravitational
fields, impacting the linear momentum, angular momentum, torque, energy, and
force and so forth.
