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I didn't want to call it an advanced introduction, at the risk of sounding absurd.

What I'm looking for is a mature and/or modern introduction to differential geometry. Let me clarify three things: what I mean by mature is that the book should assume (or at least be targeted towards) someone who has experience reading math, and say a solid background in analysis, topology, and algebra (and maybe a very basic intro to differential geometry). What I mean by modern is that it does not shy away from using modern algebraic and analytic machinery. What I mean by "and/or" is that of course, I would be interested in either modern introductions or mature introductions, due to fear that maybe both is a tall order.

I know do Carmo has "Differential Geometry of Curves and Surfaces." Is that the best choice?

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  • $\begingroup$ I recommend Clifford Taubes' book Differential Geometry: Bundles, Connections, Metrics and Curvature. $\endgroup$ – Spenser Aug 26 '18 at 9:16
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The most user-friendly and elementary introduction to Differential Geometry is Loring Tu's An Introduction to Manifolds (2nd Edition) which really tells you how to make effective calculations .
Part of the friendliness is that Tu gives at the beginning of each chapter a careful motivating explanation of its content, enriched by a short historical introduction and photograph of a main contributor.

A more comprehensive treatise which will take you further in the subject, is our friend John Lee's rich and extremely accurate Introduction to Smooth Manifolds.
It covers all the main aspects of Manifold Theory in a detailed way and has many insightful drawings, a must in a geometry book.
Do not confuse this book with Jeffrey Lee's Manifolds and Differential Geometry, which also seems excellent but which I haven't used sufficiently for me to comment on it.

The most encyclopedic treatise is Spivak's five volume (!) A Comprehensive Introduction to Differential Geometry.
I can't begin to explain why this is one othe most extraordinary books ever written in Differential Geometry and even in all of Mathematics, but let me just describe one feature.
In Volume 2 Spivak gives a 19 page translation of Riemann's inaugural lecture, which so excited Gauss, and then goes on to explain using today's notation what Riemann meant.
This is invaluable because Riemann was inventing the concept of a differentiable riemannian manifold in that talk, but didn't have the necessary tools (essentially abstract topology) to be completely rigorous.
The modern definition of a manifold was given only in 1913 by Hermann Weyl (in dimension one) and in 1936 by Hassler Whitney (in arbitrary dimension).
Similarly Spivak also explains Gauss's work in modern notation.
These "translations" from classical mathematics into modern notions and notations are just one of the gems of Spivak's masterful and incredibly original treatise.

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Although it's not exclusively targeted to a "mature audience", I think Lawrence Conlon's book Differentiable Manifolds meets your other criteria (especially about using modern algebra) and is well-written and quite detailed.

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