The author considers the problem \[ du(f,x)=({\mathcal L}u(t,x) + f(t,x))dt+ ({\mathcal N}^ l(t,x)u(t,x)+ g^ l(t,x))dW^ l(t), \quad u(0,x)=u_ 0(x), \] \(x\in M\), where \((W^{\l},{\mathcal F}_ t)\) is a \(d_ 1\)- dimensional Wiener process on a stochastic basis \((\Omega,{\mathcal F},P,({\mathcal F}_ t))\), \({\mathcal L}\) is a second order and \({\mathcal N}^ 1,\dots,{\mathcal N}^{d_ 1}\) are first order differential operators acting on differentiable functions defined on a manifold \(M\). \(g^ l\), \(f\) are adapted random fields an \(R^ +\times M\), \(u_ 0\) is an \({\mathcal F}_ 0\)-measurable random field on \(M\). The stochastic differential is defined in Itô's sense. Existence and uniqueness theorems are proved by using of well-weighted Sobolev spaces.