Want to learn differential geometry and Riemannian geometry by myself

I want to follow a Ph.D. programme on geometric analysis. The main focus are Monge-Ampere equations and convex level set of $p$-harmonic equations. The theory of PDEs is very familar to me. However, I have little knowledge of the geometry. Therefore, I want some readable texts on this issue. Since (linear) functional analysis and nonlinear analysis are my stong points, I want the texts to describe the geometry in terms of analysis.

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