View source for Two-point tensor

A tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945801.png" /> which depends on a pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945802.png" /> in a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945803.png" />, i.e. a tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945804.png" /> defined on the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945805.png" />. As an example, covariant derivatives of the [[World function|world function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945806.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945807.png" /> and its derivatives as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945808.png" />, such as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945809.png" /></td> </tr></table>

are employed in the calculus of variations and in the theory of relativity.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Synge,   "Relativity: the general theory" , North-Holland &amp; Interscience  (1960)</TD></TR></table>