Two-point tensor

A tensor $T$ which depends on a pair of points $x,x'$ in a manifold $X$, i.e. a tensor field $T(x,x')$ defined on the product $X \times X$. As an example, covariant derivatives of the world function $\Omega(x,x')$ and, in general, of an arbitrary invariant depending on two points are two-point tensors. The properties of such a tensor, in particular the limits of $T$ and its derivatives as $x' \rightarrow x$, such as $$ [T_{ij'}] =\lim_{x' \rightarrow x} \nabla_i \nabla_{j'} T(x,x') $$ are employed in the calculus of variations and in the theory of relativity.

References

[1]	J.L. Synge, "Relativity: the general theory" , North-Holland & Interscience (1960) Zbl 0090.18504