See: String theory
Investigation: Understand what the properties of a Calabi-Yau Manifold have to do with conceptual structure.
弦理论
- What is the significance of the properties of a Calabi-Yau Manifold?
A Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure.
A Calabi-Yau manifold is a compact Kähler manifold with a vanishing first Chern class, that is also Ricci flat.
A Calabi-Yau manifold thus has:
- complex structure: transition maps are holomorphic
- Riemannian structure: positive definite inner product on the tangent space at each point
- symplectic manifold: equipped with the symplectic form
- compact: "Kähler package" is a collection of further restrictions on the cohomology of compact Kähler manifolds, building on Hodge theory
- vanishing first Chern class
- Ricci flat metric Riemannian manifolds whose Ricci curvature tensor vanishes. Since Ricci curvature measures the amount by which the volume of a small geodesic ball deviates from the volume of a ball in Euclidean space, small geodesic balls will have no volume deviation, but their "shape" may vary from the shape of the standard ball in Euclidean space. For example, in a Ricci-flat manifold, a circle in Euclidean space may be deformed into an ellipse with equal area. This is due to Weyl curvature.
- In physics, Ricci-flat manifolds represent vacuum solutions to the analogues of Einstein's equations for Riemannian manifolds of any dimension, with vanishing cosmological constant.
- In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or non-gravitational fields are present.