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Results tagged with riemannian-geometry
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user 13972
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.
82
votes
Accepted
When can a connection Induce a Riemannian metric for which it is the Levi-Civita connection?
Bill and Willie have (of course) given correct answers in terms of the holonomy of the given torsion-free connection $\nabla$ on the $n$-manifold $M$. However, it should be pointed out that, practica …
- 108k
61
votes
Theorem of Bryant in higher dimensions
First, the hypotheses of the theorem I proved require that $Y$ be compact and oriented, in addition to requiring that $g$ be real-analytic.
Second, the method I used (the Cartan-Kähler Theorem) exten …
- 108k
54
votes
Does the curvature determine the metric?
It should not be surprising that, for a $2$-dimensional manifold, the Gauss curvature $K:M\to\mathbb{R}$ does not determine a unique metric $g$. After all, the former is locally one function of $2$ v …
- 108k
50
votes
What is the Levi-Civita connection trying to describe?
I think that the literal answer is that the Levi-Civita connection of $g$ is trying to describe the metric $g$ and nothing else. It is the only connection-assignment that is uniquely defined by the m …
- 108k
48
votes
Accepted
Finding a 1-form adapted to a smooth flow
If I understand correctly, there is already a counterexample on the torus:
On the $xy$-plane $\mathbb{R}^2$, let $X$ be the vector field
$$
X = \sin x\,\frac{\partial\ }{\partial x} + \cos x\,\frac{\ …
- 108k
45
votes
Riemann's formula for the metric in a normal neighborhood
Perhaps the simplest way to understand this formula is to think about how you would go about deriving it: Try to find the 'best' coordinates you can centered on a given point and see what doesn't cha …
- 108k
34
votes
Riemannian surfaces with an explicit distance function?
NB (3/1/13): I revised this answer to make it more complete (and, to be frank, more accurate). My original answer did not take into account the difference between the cut locus and the conjugate loc …
- 108k
32
votes
Accepted
Can a manifold have a curvature-free connection that is not torsion-free?
Many manifolds have curvature-free (i.e., flat) connections on their tangent bundles. For example, any orientable $3$-manifold $M$ is parallelizable, i.e., its tangent bundle is trivial, so it carrie …
- 108k
29
votes
Accepted
Torsion and parallel transport
Here is another way to think of the relation between torsion and parallel transport, one that some may find more congenial than many of the other interpretations that have been proposed:
Start with a …
- 108k
29
votes
Accepted
How should you explain parallel transport to undergraduates?
This may not reallly be an answer that you like, but I think that, maybe you misunderstood what Ben McKay was trying to describe. Here is a more explicit, extrinsic description that may help:
Suppose …
- 108k
28
votes
Do minimal submanifolds minimize area locally?
Yes, this is true, but finding an explicit general proof in the literature seems to be a challenge.
I think that many people just believe it without having seen an actual proof. One reason is that …
- 108k
26
votes
Accepted
Complex manifolds in which the exponential map is holomorphic
NB: I've had a little time to think about this and can now improve my answer, in particular, removing the real-analytic assumption, which, as I suspected, was not necessary. Here is the improved ans …
- 108k
24
votes
Accepted
What is the analog of the "Fundamental Theorem of Space Curves," for surfaces, and beyond?
I've added a few sentences to my answer to clarify something that some readers may be wondering about, which is why there isn't as simple an answer for surfaces in $3$-space as there is for curves in …
- 108k
23
votes
Accepted
Eigenfunctions of the laplacian on $\mathbb{CP}^n$
The $k$-th eigenfunctions are actually easy to describe: In $\mathbb{C}^{n+1}$ with unitary complex coordinates $z_0,z_1,\ldots,z_n$, write $Z = |z_0|^2+\cdots+|z_n|^2$.
Now, for a given $k\ge0$, …
- 108k
23
votes
Accepted
Characterizing Hessians among symmetric bilinear tensors
There are local conditions, but they typically involve the curvature tensor of the underlying metric. For example, if the metric is flat, so that one can choose orthonormal coordinates $x_i$ in which …
- 108k