Bitensor
In differential geometry and general relativity, a bitensor is a tensorial object that depends on two points in a manifold, as opposed to ordinary tensors which depend on a single point. Bitensors provide a framework for describing relationships between different points in spacetime and are used in the study of various phenomena in curved spacetime.
Definition
A bitensor is a tensorial object that depends on two points in a manifold, rather than on a single point as ordinary tensors do.A bitensor field can be formally defined as a map from the product manifold to an appropriate vector space, where is a smooth manifold and is the vector space corresponding to the tensor space being considered.
In the language of fiber bundles, a bitensor of type is defined as a section of the exterior tensor product bundle, where denotes the tensor bundle of rank and represents the exterior tensor product, where denotes the space of sections.
The exterior tensor product bundle is constructed as where are projection operators that project onto the respective factors of the product manifold, and denotes the pullback of the respective bundles.
In coordinate notation, a bitensor with components has indices associated with two different points and in the manifold. By convention, unprimed indices refer to the first point, while primed indices refer to the second point. The simplest example of a bitensor is a biscalar field, which is a scalar function of two points. Applications include parallel transport, heat kernels, and various Green's functions employed in quantum field theory in curved spacetime.