>>12838429Lee's Smooth manifolds for similar content but far far more detailed
Lee's Riemannian manifolds for "classical" modern Riemannian geometry
Tu's Differential geometry for Riemannian geometry from the principal bundles point of view
Guillemin and Pollack's Differential Topology or Milnor's Differential Topology for, well, differential topology which serves as a bridge between geometry and algebraic topology
Hatcher or Bredon for algebraic topology (not manifold stuff)
I wouldn't start with Bott, Tu unless you know some topology already
