I'm reading Jost's Riemannian Geometry and Geometric Analysis and in section 1.6 during the proof of the first step in theorem 1.6.1 it says
Step 1 is a general result from the theory of partial differential equations which follows by linearizing the equation at t = 0 and applying the implicit function theorem in Banach spaces, see §A.3. Therefore, we shall not discuss this here any further.
In appendix A.3
What one can deduce from Theorem A.3.2, however, is the short time existence of solutions when the linearization of the differential operator satisfies the assumptions of that theorem. This follows by linearization and the implicit function theorem.
Could someone provide me a resource for a more detailed way to linearize equation 1.6.2 ($\Gamma_{jk}^i u_s^ju_s^k$)?
For reference, the theorem and related parts are