Geodesic curvature in hyperbolic geometry

How is geodesic curvature defined with an ODE or geometric construction in the hyperbolic plane?

How do hyperbolic geodesics change when there is deviation from geodesy? How to construct or graph differently bent arcs in the hyperbolic plane, say in the Poincare’s disk model?

Such definition is needed if a circumcircle needs to be drawn around three vertices of a triangle with three intersecting geodesics, or if a single in-circle has to drawn touching the three.

Mean curvature flow for arcs

Do you know a reference about mean curvature flow for curves in the plane with fixed Dirichlet boundary conditions?

I start with a curve connecting two points and I let it evolve by mean curvature flow keeping the endpoints fixed. Assuming the original curve is smooth, I guess the flow should exist for every time and in the limit it should converge to the straight line segment between the endpoints. Is this correct? Can I weaken the assumptions on the starting curve (say only Lipschitz) and get the same result?

Do you know if anything has been done for general Riemannian manifolds?

Positive Ricci curvature on biquotients

I am working with biquotients and positive curvatures and I was able to give a relatively simple proof for the following:

Theorem: Let $ G$ be a compact connected Lie group with a bi-invariant metric $ Q$ . Let $ H\times K\subset G\times G$ where $ H,K$ are closed subgroups of $ G$ . Consider the action $ $ (h,k)\cdot g := hgk^{-1}$ $ and assume it is free. Then, if $ G$ is semi-simple the quotient $ H\setminus G/K$ has positive Ricci curvature on the submersion metric.

The proof uses some properties of semi-simple Lie groups such as the its decomposition as direct sums of ideals and the fact it is centerless.

My questions are: is this result stated in this way well known in the literature? Furthermore, how is the hypothesis of $ G$ being semi-simple related to Lorenz Schwachhoefer, Wilderich Tuschmann theorems on positive Ricci curvature of quotients where positive Ricci curvature is ensured (if only if condition) with $ H\setminus G/K$ having finite fundamental group?

Radius of curvature question

Im trying to figure out an equation from geometric geodesy. But doing the derivation of that equation results in a different equation then the one from the literature. Its the (1-e^2). I constantly get (e^2-1). What am I doing wrong. The calculation is in the picture.

Calculation I did and the one from the literature

Subdividing a Compact Bounded Curvature Manifold into Charts with Bounded Lipschitz Constant

Let $ M \subset \mathbb{R}^d$ be a compact smooth $ k$ -dimensional manifold embedded in $ \mathbb{R}^d$ . Let $ \mathcal{N}(\epsilon)$ denote the size of the minimum $ \epsilon$ cover $ P$ of $ M$ ; that is for every point $ x \in M$ there exists a $ p \in P$ such that $ \| x – p\|_{2}$ . My goal is to show that $ \mathcal{N}(\epsilon) \in \Theta\left(\frac{1}{\epsilon^k}\right)$ . Note that the bound depends on the dimension $ k$ , not the embedding space $ d$ .

To this end, I’d like to construct an $ \epsilon$ -cover in each coordinate chart $ (U, \phi)$ of $ M$ , where $ \phi: U \subset \mathbb{R}^k \rightarrow M$ . A cover in $ U$ can be transformed under $ \phi$ into a cover in $ M$ . If $ \phi$ is $ L$ -Lipschitz, that is $ \|\phi(x) – \phi(y)\|_2 \leq L \|x – y\|_{2}$ , then I can create an $ \epsilon$ -cover over a subset $ \phi(U) \subset M$ , with slightly larger balls than those in $ U \subset \mathbb{R}^k$ as determined by the Lipschitz constant $ L$ .

To have each coordinate chart $ (U, \phi)$ be $ L$ -Lipschitz I’m willing to assume $ M$ has bounded curvature. Can I subdivided $ M$ into a set of coordinate charts which are $ L$ -Lipschitz and how many such charts do you need as a function of the curvature of $ M$ ?

A metric has positive sectional curvature if and only if ${\rm Ric}_{ij} < \frac{r}{2}g_{ij}$

This is a cross-post from my question on MSE.

It is well known that

In dimension three a metric has positive sectional curvature if and only if $ {\rm Ric}_{ij} < \frac{r}{2}g_{ij}$ . where $ r$ denotes the scalar curvature.

Is there a higher dimensional analogues of this theorem? Any reference or counterexample?

Injectivity radius on complete manifolds with postive and bounded curvature

I have two question:

1) Are there any examples of complete manifold with strictly postive and bounded section curvature which has zero injective radius?

2) Is there a sequence of non-compact complete manifolds with strictly postive and bounded section curvature with injiective radius approch to zero?

I think one may consturct these examples from beger’s spheres, but I cannot do it rigiously.

Ricci Curvature on Grassmannian

Suppose $ G_r(n)$ is the Grassmannian, which is the collection of all $ r$ dimensional subspace in $ \mathbb{R}^{n}$ equipped with the usual invariant metric. Let $ Ricc(G_r(n))$ be the Ricci curvature tensor. What are the best known constants $ 0<c_{n,r}<C_{n,r}$ such that $ $ c_{n,r}\leq Ricc(G_r(n))\leq C_{n,r}$ $ ? I can only find results about complex Grassmannian, but not for real Grassmannian.

Curvature forms as exterior covariant derivative?

I have read on several forums like this one, that given a connection form $ \omega$ on a principal bundle and its curvature form $ \Omega$ , I can state that $ \Omega=d_\omega\omega$ alike I do in the case of torsion, namely that torsion is the covariant differential of a differential form. To me that does not make sense, because $ \Omega=d\omega+\frac{1}{2}[\omega,\omega]$ and there is no representation in which that $ \frac{1}{2}$ can arise.

Any explanations?