# Questions tagged [curvature]

In differential geometry, the term curvature tensor may refer to the Riemann curvature tensor of a Riemannian manifold, the curvature of an affine connection or covariant derivative (on tensors), or the curvature form of an Ehresmann connection. (Def: http://en.m.wikipedia.org/wiki/Curvature_tensor)

1,797 questions
Filter by
Sorted by
Tagged with
36 views

### Does any compact boundary become a minimal hypersurface iff each of the boundary component is a minimal hypersurface?

There are researchers who made the following assumptions in their paper on the Penrose inequality: Let $M$ be a complete, connected Riemannian $3$-manifold and suppose that the boundary $\partial M$ ...
• 1,149
44 views

• 489
1 vote
22 views

### Integrated Curvature on Manifolds in SageMath

Is it possible to integrate the Ricci curvature scalar on an arbitrary $n$-dimensional manifold in SageMath? Just straight up: $$\int R dV$$ Assume that I know how to get the Scalar curvature a la ...
• 2,575
23 views

33 views

### Is there any natural way to turn flat space with torsion into curved space without it?

Say we have a smooth function $f: \mathbb{R}^n \rightarrow SO(n)$, whose value is asymptotically zero in all directions. In other words, we're specifying a rotation at every point of space, but ...
• 189
32 views

• 6,450
52 views

• 277
50 views

1 vote
40 views

### Why non zero wedge term in curvature form?

The definition of the curvature form ( from Wikipedia ) is: $$\Omega = d \omega + \frac{1}{2} [ \omega \wedge \omega ]$$ I am puzzled about the rightmost term. According to what I have learned so ...
• 113
40 views

### Gauss Equation in Coordinates

Consider a $(d+1)$-dimensional pseudo-Riemannian manifold $(\mathcal{M},g)$ and a Riemannian hypersurface $(\Sigma,h)$, i.e. $\Sigma$ is an embedded submanifold of $\mathcal{M}$ of codimension $1$ ...
• 1,992
103 views

### Ito drift term of extrinsic Brownian motion aka Mean curvature of intersection of hypersurfaces

Suppose we have manifold in the form $M=f^{-1}(\{\vec{0}\})$, where $f:\mathbb{R}^d\to \mathbb{R}^p$ where $p=D-d$, and $f \in C^{\infty}$ ands its Jacobian, $J_f(x)$ has full rank on $M$. Here, we ...
• 1,763
Consider the plane $\mathbb{R}^2$ together with a metric $\mathsf{g}$. Choosing suitable local coordinates $(x_1,x_2)$, $\mathsf{g}$ takes the form :  \mathsf{g}=r(x_1,x_2)\left((dx_1)^2+\epsilon (...
I'm trying to give an example of a compact surface $S\subset\mathbb{R}^3$ with constant zero Gaussian curvature for all $p\in S$. I know that a plane would have constant zero Gaussian curvature, but ...