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Results tagged with dg.differential-geometry
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user 9471
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
4
votes
Covariant derivative of determinant of the metric tensor
The determinant of a metric makes perfectly good sense, but it is not a function, rather a $2$-density. Formally, this means that it transforms as a section of the bundle associated with the frame bun …
- 2,139
3
votes
Reading list for basic differential geometry?
S. Sternberg's recent Curvature in Mathematics and Physics is at about the same level as some of the other suggestions, but includes some extra material hard to find in textbooks at this level. It's a …
22
votes
Why is there no symplectic version of spectral geometry?
From a certain point of view the premise of the question is wrong. The study of sympletic manifolds with no additional structure is akin to differential topology rather than differential geometry. Fro …
- 2,139
4
votes
Motivations for the study of dual connections
One motivation for studying dual (also called conjugate) connections in this sense comes from the study of the equiaffine geometry of a hypersurface.
Let $i:\Sigma \to \mathbb{A}$ be a nondegenerate …
- 2,139
6
votes
Accepted
General description of surface with zero gaussian curvature
A surface of Gaussian curvature zero is locally isometric to the plane, and is said to be developable. A complete surface of Gaussian curvature zero in Euclidean three space is a cylinder (where a cyl …
- 2,139
30
votes
Accepted
When is a closed differential form harmonic relative to some metric?
A closed $k$-form is called intrinsically harmonic if there is some Riemannian metric with respect to which it is harmonic. E. Calabi (Calabi, Eugenio, An intrinsic characterization of harmonic one-fo …
- 2,139
12
votes
Accepted
"Famous" 2d Riemannian manifolds with non-constant curvature
A Killing field is preserved by the Ricci flow. By a theorem of Daskalopoulos, Hamilton and Sesum (arXiv:0902.1158), on a compact surface an ancient (defined for all negative time) solution to the Ric …
- 2,139
3
votes
Conformal structure determined by principal curvatures
When defined as a tensor taking values in the normal bundle the second fundamental form of a hypersurface in $\mathbb{R}^{n+1}$ depends only on the affine structure on the ambient space, not on its me …
- 2,139
10
votes
"Noncommutative heat equation" -- a strange generalization of Killing vectors for a flat metric
Here is an observation which does not answer the question, but does at least tell where not to look for examples. It is shown that for compact Riemannian manifolds with non-positive Ricci curvature th …
- 2,139