Bitensor

A bitensor is a generalization of a tensor that depends on two points of a differentiable manifold. On a smooth manifold M, a bitensor of type (p,q; r,s) is a tensor on the product manifold M × M that transforms as a tensor of type (p,q) at the first point x and as a tensor of type (r,s) at the second point x′. In practice, a bitensor encodes tensorial information that relates data at two distinct spacetime or manifold points.

Equivalently, a bitensor can be viewed as a section of the tensor product of pullback bundles over M × M. Concretely, it belongs to the bundle (T_x M)^{⊗ p} ⊗ (T^*_x M)^{⊗ q} ⊗ (T_{x′} M)^{⊗ r} ⊗ (T^*_{x′} M)^{⊗ s} pulled back along the projections to M × M. This two-point nature allows the bitensor to link geometric data at separate points and to transform correctly under coordinate changes at each point.

As a generalization, a bitensor reduces to an ordinary tensor if all indices refer to the same point or if one of the point arguments is fixed. Common examples include the parallel propagator, a (1,1; 0,0) bitensor that transports vectors along a geodesic, and the world function-based constructions used in Hadamard expansions.

Applications of bitensors appear throughout differential geometry and theoretical physics, notably in Green’s functions and propagators for fields in curved spacetime. They provide a formal framework for expressing two-point relations such as correlations, parallel transport, and distance measures in a coordinate-independent way. The concept is well established in the literature on general relativity and quantum field theory on curved backgrounds. See also tensor, bi-tensor, Hadamard expansion, and Green’s function.