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In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas [1]) are fundamental formulas that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold.
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle, length of curves, surface area and volume.
Ahlfors, Lars V. (1961), "Some remarks on Teichmüller's space of Riemann surfaces", Annals of Mathematics, Second Series, 74 (1): 171– 191, doi:10.2307/1970309 ...
Let be a smooth manifold and let be a one-parameter family of Riemannian or pseudo-Riemannian metrics. Suppose that it is a differentiable family in the sense that for any smooth coordinate chart, the derivatives v i j = ∂ ∂ t ( ( g t ) i j ) {\displaystyle v_{ij}={\frac {\partial }{\partial t}}{\big (}(g_{t})_{ij}{\big )}} exist and are ...
Riemannian Geometry (PDF). Princeton: Princeton University Press. OCLC 5836010. Eisenhart, Luther Pfahler (1939). Coordinate Geometry. Dover Publishing. [7] Eisenhart, Luther Pfahler (1927). Non-Riemannian geometry (PDF). New York: American Mathematical Society. [8] Eisenhart, Luther Pfahler (1909). A treatise on the differential geometry of ...
The fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique affine connection that is torsion-free and metric-compatible, called the Levi-Civita connection or (pseudo-) Riemannian connection of the given metric.
In Riemannian geometry, Gauss's lemma asserts that any sufficiently small sphere centered at a point in a Riemannian manifold is perpendicular to every geodesic through the point. More formally, let M be a Riemannian manifold, equipped with its Levi-Civita connection, and p a point of M. The exponential map is a mapping from the tangent space ...
In Riemannian geometry, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains. Let M , N {\displaystyle \scriptstyle M,\,N} be smooth Riemannian manifolds of respective dimensions m ≥ n {\displaystyle \scriptstyle m\,\geq \,n} .