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- Similarly, one can have groupoids with a compatible Riemannian metric, or complex structure, etc.
- The family "g'p" of inner products is called a Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds.
- Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion.
- The introduction of a Riemannian metric or a symplectic form gives rise to a natural isomorphism between the tangent space and the cotangent space at a point, associating to any tangent covector a canonical tangent vector.
- The divergence can also be defined on a Riemannian manifold, that is, a manifold with a Riemannian metric that measures the length of vectors.
- Formally, Gaussian curvature only depends on the Riemannian metric of the surface. This is Gauss's celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking.
- The main idea here is to prove that the existence of a non-constant polynomial "p"("z") without zeros implies the existence of a flat Riemannian metric over the sphere S2.
- A partition of unity can be used to show the existence of a Riemannian metric on an arbitrary manifold.
- The fundamental objects are called the Riemannian metric and the Riemann curvature tensor.
- Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis.
- for a Riemannian metric, and [...] for the Christoffel symbols, both coming from "G" in "Gravitation".
- The main theme of the book is a study of a moduli space of magnetic monopoles; this has a natural Riemannian metric, and a key point is that this metric is complete and hyperkähler.
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