Differential Geometry

Job: Define smooth manifolds, Riemannian metrics, and curvature tensors; state geodesic and Gauss–Bonnet-level structure used in geometry and GR.

Core

Smooth manifold \(M^n\): second-countable Hausdorff space with \(C^\infty\) atlas, transition maps \(C^\infty\). Tangent space \(T_pM\); tangent bundle \(TM\). Smooth maps \(f:M\to N\) induce \(df_p:T_pM\to T_{f(p)}N\).

Vector fields \(\mathfrak{X}(M)\); Lie bracket \([X,Y]\). Differential forms \(\Omega^k(M)\); exterior derivative \(d\), \(d^2=0\); de Rham cohomology.

Riemannian metric \(g\): smooth positive-definite inner product on each \(T_pM\). Length \(L(\gamma)=\int_a^b\|\gamma'(t)\|\,dt\); energy \(E(\gamma)=\frac12\int\|\gamma'\|^2\). Levi-Civita connection \(\nabla\): unique torsion-free metric-compatible affine connection (Koszul formula).

Geodesics: \(\nabla_{\dot\gamma}\dot\gamma=0\). Exponential map \(\exp_p:T_pM\supset U\to M\). Completeness (Hopf–Rinow): metric completeness \(\Leftrightarrow\) geodesic completeness \(\Leftrightarrow\) closed bounded sets compact (connected Riemannian).

Curvature tensors (sketch). - Riemann: \(R(X,Y)Z=\nabla_X\nabla_Y Z-\nabla_Y\nabla_X Z-\nabla_{[X,Y]}Z\).
Components \(R^\rho_{\ \sigma\mu\nu}\) in coordinates; symmetries \(R_{\rho\sigma\mu\nu}=-R_{\sigma\rho\mu\nu}=\ldots\) (first Bianchi if torsion-free). - Sectional curvature of plane \(\sigma=\mathrm{span}\{u,v\}\): $$ K(\sigma)=\frac{\langle R(u,v)v,u\rangle}{|u|^2|v|^2-\langle u,v\rangle^2}. $$ - Ricci: \(\mathrm{Ric}(Y,Z)=\mathrm{tr}(X\mapsto R(X,Y)Z)\); scalar \(S=\mathrm{tr}_g\mathrm{Ric}\). - Einstein: \(\mathrm{Ric}=\lambda g\). Vacuum Einstein equation \(\mathrm{Ric}=0\) (Lorentzian signature in GR; Riemannian analogue is Ricci-flat).

Constant curvature models: \(S^n\) (\(K=1\)), \(\mathbb{R}^n\) (\(K=0\)), \(H^n\) (\(K=-1\)) — space forms.

Gauss–Bonnet (compact oriented surface): \(\int_M K\,dA=2\pi\chi(M)\). Higher-D Chern–Gauss–Bonnet pairs Pfaffian of curvature with Euler characteristic.

Submanifolds: second fundamental form \(\mathrm{II}\); Gauss equation relates ambient \(R\) to intrinsic \(R\) + \(\mathrm{II}\) terms. Theorema egregium: \(K\) intrinsic for surfaces.

See it

Unit sphere \(S^2\subset\mathbb{R}^3\): geodesics are great circles; \(K\equiv1\); \(\int K=4\pi=2\pi\chi(S^2)\). Parallel transport around a spherical triangle rotates by angle excess — holonomy = curvature.

Check

Why is torsion-free needed for the algebraic symmetries of \(R\) used in sectional curvature?

First Bianchi identity \(R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0\) uses zero torsion. Sectional curvature from the \((0,4)\) tensor needs the standard skew symmetries guaranteed for Levi-Civita.

Does \(\mathrm{Ric}=0\) imply flatness?

No in \(\dim\ge4\) (e.g. Calabi–Yau Ricci-flat Kähler metrics with nontrivial \(R\)). In \(\dim=2\), \(\mathrm{Ric}=\frac12 S g\) and \(S=2K\), so Ricci-flat \(\Rightarrow\) flat.

Pitfalls

  • Connection coefficients \(\Gamma\) are not tensorial; \(R\) and torsion \(T\) are.
  • Lorentzian geometry (GR): Hopf–Rinow fails; causal structure dominates.
  • Coordinate curvature formulas are error-prone; prefer Cartan structure equations / moving frames when possible.

Refs

Lee, Introduction to Smooth Manifolds / Riemannian Manifolds; do Carmo; Spivak I–V; Petersen; O'Neill (semi-Riemannian).

Next

Non-Euclidean geometry · Topology · Multivariable calculus · Higher Teichmüller theory — hyperbolic geometry / symmetric spaces as the geometric setting · NCG / spectral triples — Connes distance formula recovers geodesic distance; reconstruction theorem