Articles of curvature

how to find the maximum height perpendicular to arc without knowing radius of the arc

I am doing the curvature correction of the object boundary, I need to calcluate the maximum height perpendicular to the arc shown in the figure. I know the chord length and arc length, But i don’t know the radius of the arc. I would like to know how to find out the maximum height perpendicular […]

Visual explanation of the indices of the Riemann curvature tensor

I’m trying to understand the meaning of the Riemann curvature tensor, but I don’t seem to be ready yet to understand the detailed rigorous definition. Anyway, I managed to understand (this gif was really helpful) that it takes two coordinate lines $x^\mu, x^\nu$, parallel transports a vector first on $x^\mu$ and then $x^\nu$ and compares […]

maximum curvature of 2D Cubic Bezier

Given a 2D cubic Bezier segment defined by $P_0, P_1, P_2, P_3$, here’s what I want: A function that takes the segment and outputs the maximum curvature without using an iterative approach. I have a function that finds the maximum curvature at the moment, but does this using Brent’s Method to search a range of […]

Injectivity radius of Exponential and curvature

Define : Injectivity radius , Exponential This question is considered in Riemann manifold. I think the Injectivity radius is connect with curvature. I guess the Injectivity radius can be controlled by curvature.I think there should be some function make the below inequality right. $$ f(curvature)\leq \text{Injectivity radius} \leq g(curvature) $$ Is it right? Or there […]

Parallel vector fields imply a flat connection in constant curvature

This a continuation of a previous question, which you can find here. Question: Let $M$ be a Riemann surface with constant curvature, but assume $M$ is not flat. In fact, I would be satisfied with an analysis of the case $M = S^2$. Can there exist two pointwise linearly-independent vector fields $X$ and $Y$ defined […]

Can someone explain curvature in simple terms

I am studying differential geometry but am having a hard time picturing curvature. Can anyone explain it to me in simple terms, perhaps with any diagrams. As simple as possible!

Riemann tensor in terms of the metric tensor

The Riemann tensor is a function of the metric tensor when the Levi-Civita connection is used. Most tensor notation based texts give the Riemann tensor in terms of the Christoffel symbols, which are give in terms of the partial derivatives of the metric, but I have not seen the Riemann tensor given directly in terms […]

Ricci SCALAR curvature

Are there any manifolds that have NULL SCALAR curvature but not null ricci curvature tensor? For dim>2, obviously. Edit: Are there, also, any manifolds of null scalar curvature but ricci curvature not proportional to the metric tensor?

parallel postulate of Euclidean geometry and curvature

In elementary geometry, we have two standard examples which violate the (strong) parallel postulate of Euclidean geometry: in hyperbolic geometry, we have more than one parallel through a point which doesn’t lay on a given line, and in spherical geometry, we have no parallels at all. In both of these geometries, we have some kind […]

Embedding manifolds of constant curvature in manifolds of other curvatures

I know that there is no complete surface embedded in $\mathbb{R}^3$ of constant curvature -$k$ for any $k$. But you can clearly embed the hyperbolic plane (curvature -1) into hyperbolic 3-space (curvature -1). You can also embed Euclidean planes (curvature 0) as horoballs into hyperbolic 3-space (curvature -1) But can you embed surfaces of curvature […]