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Riemannian Geometry

Field: Geometry

The study of Riemannian manifolds, smooth manifolds with a Riemannian metric.

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Bernhard Riemann Riemannian geometry was first put forward in generality by Bernhard Riemann in the 19th century. It deals with a broad range of geometries whose metric properties vary from point to point, including the standard types of non-Euclidean geometry. The main idea is that a space, like a surface in Euclidean space, carries what is known as a Riemannian metric, which arises by restricting the ambient dot product to vectors that are tangent to the surface. Riemann realized that the essential ingredient here was this quadratic form on tangent vectors, and that it could be generalized. The important thing, the intrinsic way that paths in the surface could be measured, was not how the surface sat in space, but how this quadratic form varied from point to point. Consider the simple case of a cylinder: a flat piece of paper can be wrapped into a cylinder, but the "intrinsic distance", that is the distance that an insect must crawl to get from one point to another, is not changed by the warping of a flat paper into three dimensions. A more advanced example, known to Riemann, was that the helicoid could (after cutting along a generator) be deformed to a catenoid without altering the intrinsic geometry (what an ant sees). Deformation of a right-handed helicoid into a left-handed one and back again via a catenoid, preserving this intrinsic metric (length of sides of mesh) Riemann's idea was that it was the quadratic form which matters most, rather than the particular way a surface might be realized in space (a cylinder versus a piece of paper, for example). Riemannian geometry thus studies the intrinsic geometry of a manifold, equipped with a quadratic form on tangent vectors at every point. An important idea is that manifolds, unlike surfaces, need not be described as embedded in any particular Euclidean space: they may be described in local coordinate patches. In each coordinate patch, the metric has one expression, and when going to another patch, the metric changes by well-defined rules (essentially the chain rule). A modern theorem is that everysmooth manifold admits a Riemannian metric (in fact, many Riemannian metrics). The properties of such metrics are useful to constrain the topology of the original manifold. In Riemannian geometry, as in Euclidean geometry, the quadratic form is positive definite. Relaxing this condition, and allowing that some non-zero vectors can be null under the quadratic form allows the structure of pseudo-Riemannian manifolds, which (in four dimensions) are the main objects of the theory of general relativity. On the other hand, replacing the quadratic form by a more general non-quadratic function leads to Finsler geometry. There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals. Dislocations and disclinations produce torsions and curvature. The following articles provide some useful introductory material: - Metric tensor - Riemannian manifold - Levi-Civita connection - Curvature - Riemann curvature tensor - List of differential geometry topics - Glossary of Riemannian and metric geometry - ^Kleinert, Hagen (1989), Gauge Fields in Condensed Matter Vol II, World Scientific, pp. 743–1440, archived from the original on 2022-08-22, retrieved 2011-07-17 - ^Kleinert, Hagen (2008), Multivalued Fields in Condensed Matter, Electromagnetism, and Gravitation(PDF), World Scientific, pp. 1–496, Bibcode:2008mfcm.book.....K, archived from the original(PDF) on 2022-01-20, retrieved 2011-07-17
Let (M,g)(M, g) be a complete Riemannian manifold, where MM is a smooth manifold and gg is a Riemannian metric tensor on MM. Let p be a fixed point in Mp \text{ be a fixed point in } M, and TpMT_p M be the tangent space at pp. The exponential map expp:TpM→M\text{exp}_p: T_p M \to M is defined by mapping a tangent vector v to the point reached by the unique geodesic expp(v)=geod(t)∣t=1 starting at p with initial velocity vv \text{ to the point reached by the unique geodesic } \text{exp}_p(v) = \text{geod}(t)|_{t=1} \text{ starting at } p \text{ with initial velocity } v. Classical theorems often establish conditions under which expp\text{exp}_p is a local diffeomorphism, or when the manifold is globally isometric to a simpler space. Specifically, the Cartan-Hadamard theorem states that if (M,g)(M, g) is a complete, simply connected manifold with non-positive sectional curvature K (i.e., K is negative or zero everywhere),K \text{ (i.e., } K \text{ is negative or zero everywhere)}, then the exponential map expp:TpM→M\text{exp}_p: T_p M \to M is a global diffeomorphism, implying that (M,g)(M, g) is globally isometric to a Hadamard manifold, which is a simply connected, complete manifold with non-positive sectional curvature.
- Gauss–Bonnet theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ(M) where χ(M) denotes the Euler characteristic of M. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem. - Nash embedding theorems. They state that every Riemannian manifold can be isometrically embedded in a Euclidean spaceR^{n}.
Let MM be an nn-dimensional smooth manifold, equipped with an atlas A={(Ui,ϕi)}i=1k\text{A} = \{\text{(U}_i, \phi_i\text{)}\text{}\}_{i=1}^{k}. A Riemannian metric gg on MM is a smooth section of the bundle Sym2(T∗M)\text{Sym}^2(T^*M), which assigns to every point p∈Mp \in M a positive-definite inner product gpg_p on the tangent space TpMT_p M. Formally, gg must satisfy the following conditions:\n\n1. **Smoothness:** For any local chart ϕi:Ui→Rn\phi_i: U_i \to \mathbb{R}^n, the components gij=g(∂∂xi,∂∂xj)g_{ij} = g(\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}) must be smooth functions on UiU_i.\n2. **Positive Definiteness:** For all p∈Mp \in M and any non-zero tangent vector v∈TpMv \in T_p M, the metric must satisfy gp(v,v)>0g_p(v, v) > 0.\n\nThus, a Riemannian manifold is the pair (M,g)(M, g), where gg is a smooth, positive-definite, symmetric (0,2)(0, 2)-tensor field on MM. The associated Riemannian volume element is volg=det⁡(gij) dx1∧⋯∧dxn\text{vol}_g = \sqrt{\det(g_{ij})}\ dx^1 \wedge \dots \wedge dx^n.