53-XX	Differential geometry {For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx}
53-00	General reference works (handbooks, dictionaries, bibliographies, etc.)
53-01	Instructional exposition (textbooks, tutorial papers, etc.)
53-02	Research exposition (monographs, survey articles)
53-03	Historical {!must also be assigned at least one classification number from section 01}
53-04	Explicit machine computation and programs (not the theory of computation or programming)
53-06	Proceedings, conferences, collections, etc.
53Axx	Classical differential geometry
53A04	Curves in Euclidean space
53A05	Surfaces in Euclidean space
53A07	Higher-dimension and -codimension surfaces in Euclidean n-space
53A10	Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
53A15	Affine differential geometry
53A17	Kinematics
53A20	Projective differential geometry
53A25	Differential line geometry
53A30	Conformal differential geometry
53A35	Non-Euclidean differential geometry
53A40	Other special differential geometries
53A45	Vector and tensor analysis
53A55	Differential invariants (local theory), geometric objects
53A60	Geometry of webs [See also 14C21, 20N05]
53A99	None of the above, but in this section
53Bxx	Local differential geometry
53B05	Linear and affine connections
53B10	Projective connections
53B15	Other connections
53B20	Local Riemannian geometry
53B21	Methods of Riemannian geometry
53B25	Local submanifolds [See also 53C40]
53B30	Lorentz metrics, indefinite metrics
53B35	Hermitian and K\"ahlerian structures [See also 32Cxx]
53B40	Finsler spaces and generalizations (areal metrics)
53B50	Applications to physics
53B99	None of the above, but in this section
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 32Q20]
53C10	G-structures
53C12	Foliations (differential geometric aspects) [See also 57R30, 57R32]
53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C17	Sub-Riemannian geometry
53C20	Global Riemannian geometry, including pinching [See also 31C12, 58B20]
53C21	Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]
53C22	Geodesics [See also 58E10]
53C23	Global topological methods (\`a la Gromov)
53C24	Rigidity results
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C26	Hyper-K\"ahler and quaternionic K\"ahler geometry, ``special'' geometry
53C27	Spin and Spinc geometry
53C28	Twistor methods [See also 32L25]
53C29	Issues of holonomy
53C30	Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]
53C35	Symmetric spaces [See also 32M15, 57T15]
53C38	Calibrations and calibrated geometries
53C40	Global submanifolds [See also 53B25]
53C42	Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
53C43	Differential geometric aspects of harmonic maps [See also 58E20]
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
53Dxx	Symplectic geometry, contact geometry [See also 37Jxx, 70Gxx, 70Hxx]
53D05	Symplectic manifolds, general
53D10	Contact manifolds, general
53D12	Lagrangian submanifolds; Maslov index
53D15	Almost contact and almost symplectic manifolds
53D17	Poisson manifolds
53D20	Momentum maps; symplectic reduction
53D22	Canonical transformations
53D25	Geodesic flows
53D30	Symplectic structures of moduli spaces
53D35	Global theory of symplectic and contact manifolds [See also 57Rxx]
53D40	Floer homology and cohomology, symplectic aspects
53D45	Gromov - Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35]
53D50	Geometric quantization
53D55	Deformation quantization, star products
53D99	None of the above, but in this section
53Z05	Applications to physics