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Differential Geometry (Definition)

The study of geometry using the techniques of calculus and linear algebra, focusing on smooth manifolds, curvature, etc.
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The statement of the theorem

Let MM be an nn-dimensional smooth manifold. Differential Geometry is the study of geometric structures defined on MM. Formally, a Riemannian manifold is a pair (M,g)(M, g), where gg is a Riemannian metric, i.e., a smooth assignment of a positive-definite inner product gp:TpM×TpM→ℜg_p: T_p M \times T_p M \to \real to every tangent space TpMT_p M. The geometry is then characterized by the Levi-Civita connection ∇\nabla, which is the unique torsion-free connection compatible with gg (i.e., ∇g=0\nabla g = 0). The fundamental geometric invariants are derived from the Riemann curvature tensor RR, defined by the commutator of covariant derivatives: \n\nR(X,Y)Z=∇X∇YZ−∇Y∇XZ−∇[X,Y]ZR(X, Y) Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z \n\nFor any tangent vector fields X,Y,Z∈Γ(TM)X, Y, Z \in \Gamma(TM), the curvature tensor RR yields a (0,4)(0, 4)-tensor field R(X,Y,Z,W)=g(R(X,Y)Z,W)R(X, Y, Z, W) = g(R(X, Y) Z, W). The theory investigates how this tensor field encodes the intrinsic curvature of MM, allowing for the calculation of geometric quantities such as the Ricci curvature Ric(Y,Z)=Tr(X↦R(X,Y)Z)\text{Ric}(Y, Z) = \text{Tr}(X \mapsto R(X, Y) Z) and the scalar curvature S=g−1(Ric)S = g^{-1}(\text{Ric}). The study thus involves analyzing the structure group of the metric gg and the resulting curvature invariants.