40
votes

### What is the best way to draw curvature?

The best way I know to illustrate the notion of curvature is via Toponogov's theorem. We can compare any (geodesic) triangle in a Riemannian manifold $M$ with one with the same edge lengths in ...

32
votes

Accepted

### Is the minimal volume a topological invariant?

Minimal volume is not a homeomorphism invariant. It is shown in [L. Bessières, Un théorème de rigidité différentielle, Comm. Math. Helv. 73 443-479 (1998)] that the minimal volume of the connected sum ...

19
votes

Accepted

### Techniques to solve a non-linear differential equation related to curvature

Well the standard techniques would take advantage of the fact that the equation doesn't explicitly involve the independent variable $x$ to integrate the equation once, thereby leading to the ...

18
votes

Accepted

### Geometric interpretation of the Weyl tensor?

There is such an interpretation, with a few caveats. Essentially, there is a canonical connection on a certain vector bundle for which the "principal part" of the curvature is the Weyl ...

17
votes

Accepted

### Constant Gaussian curvature surfaces in 3-space containing lines

Given any point $p$ on a surface $S$ of Gauss curvature -1,
there exists an open neighborhood $U\subset S$ and $p$-centered
coordinates $(x,y):U\to\mathbb{R}^2$, whose image is
a domain $R = (x,y)(U)\...

17
votes

### What is the best way to draw curvature?

With advances in discrete differential geometry, it is now nearly
routine to compute curvature on meshed surfaces. Here are two
of many possible color-coded examples.
...

16
votes

Accepted

### Are there some intrinsic invariants of surfaces other than Gaussian curvature?

As others have pointed out, it's not hard to show that any function $F(\kappa_1,\kappa_2)$ that is intrinsic to the surface metric must be a function of $K = \kappa_1\kappa_2$, so that settles what ...

16
votes

### What is the best way to draw curvature?

This is not what you're looking for, but I always remember Milnor's diagram in Chapter 9 of his book on Morse Theory describing the symmetries of the curvature tensor.

Community wiki

15
votes

Accepted

### Locally Riemannian Connection

Perhaps I can offer some information and comment on this problem. An essential part of the problem is how to interpret terms such as 'observe', 'accessible', 'identify', as the OP wants to know how ...

15
votes

### Does every ‘curvature’ tensor induce a metric?

Here's a quick summary. The answers provided in the link cited by @RBega2 have more details.
Given a curvature-like tensor $R$ at a point, there always exists a metric whose curvature tensor at that ...

14
votes

Accepted

### Taylor expansion of the metric tensor in the normal coordinates

Using the reference https://arxiv.org/pdf/0903.2087.pdf, which agrees with https://arxiv.org/pdf/hep-th/0001078v1.pdf, which agrees with the reference U. Müller, C. Schubert and Anton M. E. van de Ven,...

14
votes

Accepted

### Gauss-Bonnet Theorem: Neither Gauss nor Bonnet

Shiing-Shen Chern's Historical remarks on Gauss-Bonnet seems an authoritative source. The formula goes back to Gauss (1827), Bonnet (1848), and Binet (unpublished). Gauss considered a triangle, Bonnet ...

13
votes

### Riemannian vs Non-Riemannian curvature

NB: In what follows, to save typing, I will be working on a manifold $M$, but I will write $T$, $T^*$, etc. to denote the bundles $TM$, $T^*M$, etc. and let $M$ be understood.
It seems that the OP ...

13
votes

### What is the best way to draw curvature?

Mohammed Ghomi's answer reminds me of a related picture that Cedric Villani drew to depict Ricci curvature ([1] Chapter 14). Similar to the $\operatorname{CAT}(\kappa)$ inequality, this idea can be ...

Community wiki

12
votes

Accepted

### Principal curvatures of $\mathbb{R}^{n^2}$-embedded SO(n)

What you are asking about is the second fundamental form of the embedding. Since this is a Lie group, it's enough to know what the second fundamental form is at the identity matrix $I_n=e$. Since ...

11
votes

### Differential geometric interpretation of cohomology

For geometric interpretations of cup product and Poincaré duality let us assume that in the following (dual) homology classes are representable by smooth submanifolds (and everything is orientable). ...

11
votes

### $S^3 \setminus S^1$ doesn't have hyperbolic structure

You probably mean $M$ does not admit complete hyperbolic metrics of finite volume.
Since $M$ is topologically the interior of a solid torus, a complete hyperbolic structure just identifies $M$ as the ...

11
votes

### Curvature of nonsymmetric metric tensors?

Consider a bilinear form $b \in \mathcal{C}^\infty (T^*M\otimes T^*M, \mathbb{R})$ and an affine connection $\nabla \colon \mathcal{\Gamma}^\infty(TM) \to \mathcal{\Gamma}^\infty(T^*M\otimes TM)$ ...

11
votes

### Taylor expansion of the square of the distance function on a Riemannian manifold

This is just a remark about an alternative, more `low tech', derivation of this famous formula by using Taylor series.
It relies on this property of a Riemannian metric: If $(M,g)$ is a Riemannian ...

10
votes

Accepted

### Who first proved that a vanishing Riemann tensor is sufficient for the existence of Euclidean coordinates?

Presumably it was Riemann himself who proved the sufficiency of $R=0$ for the flatness. Spivak's Chapter 4 in Volume II of "A comprehensive introduction to differential geometry" is a good source on ...

10
votes

Accepted

### What is known about Lie groups with (strictly) positive curvature?

The following result is, for example, exercise 3 on pg. 104 of Do Carmo's Riemannian Geometry book.
Suppose $X$ is a Killing field on a compact even dimensional Riemannian manifold of positive ...

10
votes

### $S^3 \setminus S^1$ doesn't have hyperbolic structure

The following contribution comes from conversations with Bill Goldman, any mistakes however are mine alone.
Any (geodesically complete) geometric 3-manifold $N=\mathbb{M}/G$ with infinite order ...

10
votes

Accepted

### Is every metric uniformly close to a metric with negative scalar curvature?

Jochen Lohkamp answers your first question in "Curvature h-principles", Ann. of Math, vol 142, p 457–498. If $\dim M\ge 3$, then the space of negative scalar curvature metrics is dense in the $C^0$ ...

9
votes

Accepted

### minimal surfaces in $S^n$

Without embeddedness, the Choi--Schoen theorem is false.
For example, there is a huge family of rotationally symmetric immersed tori in $\mathbb{S}^3$ (the only embedded one is the Clifford torus, ...

9
votes

Accepted

### Positive scalar curvature on the double of a manifold

The answer is negative at least if you do not add some sort of convexity hypothesis for the boundary, and at least in dimension $2$. Take a round $2$-sphere with $h\ge 2$ round holes. It has positive ...

9
votes

Accepted

### What should a meaningful notion of curvature satisfy, in the absence of a smooth structure?

A curvature bound for a non-smooth metric space is often known as a "synthetic curvature bound" or a "coarse curvature." Here's one possible definition for the concept.
Definition:...

8
votes

### Do curvature differences obstruct a.e orientation-preserving isometries?

There is a discontinuous map $f\colon\mathbb{S}^2\to\mathbb{R}^2$ such that $d_xf$ is defined and isometric for almost all $x$. (If you want a continuous one then I am sure the answer is "no")
To ...

8
votes

Accepted

### Holonomy of a Ricci-flat affine connection

The answer depends on the dimension. When $n=2$, Ricci-flatness of a connection implies that it is flat, so, in that case, yes, you get holonomy reduction locally. However, when $n>2$, Ricci-...

8
votes

Accepted

### Realizing the cross product of $\mathbb{R}^3$ as the curvature tensor of a Riemannian metric on $\mathbb{R}^3$

There are two interpretations of your question.
Metric cross product
Assuming that you are looking for a Riemannian metric $g$ such that (by the triple-product formula)
$$ R_g(X,Y)Z = g(Z,X)Y - g(...

8
votes

Accepted

### Positive scalar curvature on the total space of a circle bundle

It is a theorem of Gromov and Lawson, also Schoen and Yau, that no closed orientable three-manifold which contains an aspherical factor in its prime decomposition can admit a metric of positive scalar ...

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