### 2.14 Arbitrary surface in curved spacetime: general structure of UV divergences

The definition of the entanglement entropy and the procedure for its calculation generalize
to curved spacetime. The surface can then be any smooth closed co-dimension two
surface,
which divides the space into two sub-regions. In Section 3 we will consider in detail the case where this
surface is a black-hole horizon. Before proceeding to the black-hole case we would like to specify the general
structure of UV divergent terms in the entanglement entropy. In -dimensional curved spacetime,
entanglement entropy is presented in the form of a Laurent series with respect to the UV cutoff (for
see [204])
where is proportional to the area of the surface . All other terms in the expansion (48) can be
presented as integrals over of local quantities constructed in terms of the Riemann curvature of the
spacetime and the extrinsic curvature of the surface . Of course, the intrinsic curvature of the surface
can be expressed in terms of and using the Gauss–Codazzi equations. Since nothing should
depend on the direction of vectors normal to , the integrands in expansion (48) should be even powers
of extrinsic curvature. The general form of the term can be symbolically presented in the form
where stands for components of the Riemann tensor and their projections onto the sub-space
orthogonal to and labels the components of the extrinsic curvature. Thus, since the integrands are
even in derivatives, only terms , appear in Eq. (48). If is even, then there also
may appear a logarithmic term . The term in Eq. (48) is a UV finite term, which may
also depend on the geometry of the surface , as well as on the geometry of the spacetime
itself.