Differential forms are mathematical objects used in differential geometry to describe the geometrical properties of a space. They are a generalization of the concept of a vector, and can represent quantities such as length, area, and volume.
In general relativity, differential forms are used to describe the curvature of spacetime. They are used in the formulation of Einstein's field equations, which relate the curvature of spacetime to the distribution of matter and energy.
Differential forms and tensors are closely related mathematical objects. In fact, differential forms can be thought of as a type of tensor. They both describe geometrical properties of a space, but differential forms are better suited for certain calculations in differential geometry.
Parallel transport is a mathematical concept used to describe how vectors change as they are moved along a path in a curved space. Differential forms can be used to define a notion of parallel transport for arbitrary tensors, allowing for a more general approach to studying curvature in curved spaces.
Yes, differential forms are used in various other areas of physics, including electromagnetism, fluid mechanics, and quantum mechanics. They are a powerful tool for describing and understanding the properties of physical systems in a geometrically consistent way.