I were a Springer-Verlag Graduate Text in Mathematics, I would be Frank Warner's Foundations of Differentiable Manifolds and Lie Groups.

I give a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. I include differentiable manifolds, tensors and differentiable forms. Lie groups and homogenous spaces, integration on manifolds, and in addition provide a proof of the de Rham theorem via sheaf cohomology theory, and develop the local theory of elliptic operators culminating in a proof of the Hodge theorem. Those interested in any of the diverse areas of mathematics requiring the notion of a differentiable manifold will find me extremely useful.

Which Springer GTM would you be? The Springer GTM Test