The author proves the following theorem. (i) The unit tangent bundle \(U({\mathbb R}P^2(1))\) over the two-dimensional real projective space \({\mathbb R}P^2(1)\) is isometric to the lens space \(S^3(1/4)/\varGamma(4,1)\). (ii) The Lie algebra of the Killing vector fields on this space is given by \[ i(U({\mathbb R}P^2(1)))=\text{span}\{X^L, F; X\in i{\mathbb R}P^2(1))\}, \] where \(X^L\) denotes the natural lift of \(X\), and \(F\) denotes the geodesic spray on \(U({\mathbb R}P^2(1))\). Hence, the dimension of the lens space \(S^3(1/4)/\varGamma(4,1)\) is five. (iii) Any integral curve of the natural lift \(X^L\) is a geodesic in \(U({\mathbb R}P^2(1))\). Conversely, every geodesic in the unit tangent bundle can be obtained in this way.