53Cxx Global differential geometry [See also 51H25, 58-XX; for related bundle theory, see 55Rxx, 57Rxx]

• 53C05 Connections, general theory
• 53C07 Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills) [See also 32L07]
• 53C10 $G$-structures
• 53C12 Foliations (differential geometric aspects) [See also 57R30, 57R32]
• 53C15 General geometric structures on manifolds (almost complex, contact, symplectic, almost product structures, etc.)
• 53C20 Global Riemannian geometry, including pinching [See also 31C12, 58B20]
• 53C21 Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58G30]
• 53C22 Geodesics [See also 58E10]
• 53C23 Global topological methods (\`a la Gromov)
• 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
• 53C30 Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]
• 53C35 Symmetric spaces [See also 32M15, 57T15]
• 53C40 Global submanifolds [See also 53B25]
• 53C42 Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
• 53C45 Global surface theory (convex surfaces \`a la A. D. Aleksandrov)
• 53C50 Lorentz manifolds, manifolds with indefinite metrics
• 53C55 Hermitian and K\"ahlerian manifolds [See also 32Cxx]
• 53C56 Other complex differential geometry [See also 32Cxx]
• 53C60 Finsler spaces and generalizations (areal metrics) [See also 58B20]
• 53C65 Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]
• 53C70 Direct methods ($G$-spaces of Busemann, etc.)
• 53C75 Geometric orders, order geometry [See also 51Lxx]
• 53C80 Applications to physics
• 53C99 None of the above, but in this section