The Darboux-Sauer theorem establishes a relationship between infinitely small deformations (i.s.d.) of an arbitrary surface in Euclidean space and the i.s.d. of its image under a projective transformation of the space. Pogorelov established a similar relationship between the i.s.d. of a surface in a space of constant curvature and the i.s.d. of its image under a projective (geodesic) mapping of the space into a Euclidean space. In the present paper we show that both of these theorems are easily obtained from two simple results concerning Killing vector fields (i.s. motions of a space). The first of these results consists in the proportionality of the covariant components of such fields in spaces in geodesic correspondence (in the coordinates carried over by this correspondence). The second theorem establishes a relationship between i.s.d. and Killing fields in spaces of constant curvature, thereby generalizing the well-known construction of the field of rotations for i.s.d. in a Euclidean space.