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Results tagged with riemannian-geometry
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user 6871
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
2
votes
norm of $n$-th covariant derivative of smooth function
If $A$ is any section of a vector bundle $E$ over a smooth manifold $M$ and if $\nabla$ is any covariant derivative on $E$, then $\nabla A$ is a section of $T^*M \otimes E$, and this has a natural (po …
- 3,540
1
vote
Holonomy group of $\mathbb{O}P^1$
Of course, it depends on the Riemannian metric you put on $\mathbb O \mathbb P^1 \cong S^8$. For the round metric you do indeed get $\mathrm{SO}(8)$ holonomy. A priori, one could imagine other "natura …
- 3,540
23
votes
2
answers
6k
views
Why are they called isothermal coordinates?
On a Riemannian manifold, a coordinate system is called "isothermal" if the Riemannian metric in those coordinates is conformal to the Euclidean metric:
$$g_{ij} = e^{f} \delta_{ij}$$
My question is …
- 3,540
85
votes
Accepted
$G_2$ and Geometry
I promised Sean a detailed answer, so here it is.
As José has already mentioned, it is only $G_2$ (of the five exceptional Lie groups) which can arise as the holonomy group of a Riemannian manifold. …
- 3,540
10
votes
Accepted
Calabi - Yau Manifolds
There are several different "definitions" of Calabi-Yau manifolds, not all equivalent, and not all contained in one general definition. A good discussion of some of these inequivalent definitions can …
- 3,540
19
votes
Open questions in Riemannian geometry
There is also "Review of Geometry and Analysis" by S.-T. Yau (Asian Journal of Mathematics, vol. 4, no. 1, pp. 235-278, March 2000), where he discusses many big open problems in Riemannian geometry, s …