Articles of riemannian geometry

Geodesics (2): Is the real projective plane intended to make shortest paths unique?

From the Wikipedia article on geodesics: In Riemannian geometry geodesics are not the same as “shortest curves” between two points, though the two concepts are closely related. The difference is that geodesics are only locally the shortest distance between points […]. Going the “long way round” on a great circle between two points on a […]

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 […]

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 […]

Exponential map of Beltrami-Klein model of hyperbolic geometry

In the Betrami-Klein model of hyperbolic geometry, geodesics are represented as straight lines. Hence the exponential map of a tangent vector $\mathbf{v}$ at a point $\mathbf{p}$ is $\mathbf{p} + \lambda \mathbf{v}$, where $\lambda$ is a scalar that depends on $\mathbf{p}$ and $\mathbf{v}$. For example, suppose $\mathbf{p} = 0$. Then the exponential map is $$ \exp_\mathbf{p}(\mathbf{v}) […]

Example for non-Riemann integrable functions

According to Rudin (Principles of Mathematical Analysis) Riemann integrable functions are defined for bounded functions.For every bounded function defined on a closed interval $[a,b]$ Lower Riemann Sum and Upper Riemann sum are bounded .More mathematically $m(b-a) \leq L(P,f) \leq U(P,f) \leq M(b-a)$ where $m,M$ are lower and upper bounds of the function $f$ respectively. Rudin […]

Isometry in Hyperbolic space

Let $\mathbb{H}^2=\{ (x,y)\in\mathbb{R}|\ y>0 \}$ the hyperbolic space with the metric $g=(dx^2+dy^2)/y^2$. Let $$\psi(x,y)=(\psi_1(x,y),\psi_2(x,y))=\left(\mathrm{Re}\frac{az+b}{cz+d},\mathrm{Im}\frac{az+b}{cz+d}\right),$$ where $z=x+iy$. The numbers are chosen such that $ad-bc=1$. I want to show that $\psi$ is an isometry, that is \begin{equation} \psi^*g=g\quad (1) \end{equation}. In coordinates the above equation is \begin{equation} g_{\mu\nu}(x,y)=g_{\alpha\beta}(\psi(x,y))\frac{\partial \psi_\alpha(x,y)}{\partial x_\mu}\frac{\partial \psi_\beta(x,y)}{\partial x_\nu},\quad (2) \end{equation} where $x_1=x$ and […]

Whether Ricci flow keep the diagonal of metric and Ricci tensor?

Consider Ricci flow on a compact smooth Riemannian manifold $(M,g(t))$,the Ricci tensor is $Ric(t)$. Then , they meets $$ \partial_tg_{ij}=-2R_{ij} \\ \partial_tR_{ik}=\Delta R_{ik}+2g^{pr}g^{qs}R_{piqk}R_{rs}-2g^{pq}R_{pi}R_{qk} $$ If at $t=0$, the metric and Ricci tensor is diagonal, namely , $g_{ij}(0)=0 ~R_{ij}(0)=0~,~i\ne j$ , whether the metric and Ricci tensor keep diagonal under Ricci flow ?

What does it mean that we can diagonalize the metric tensor

On a Riemannian manifold $M$, the matrix representation is diagonalisable, cause the tensor is symmetric. What is the physical meaning behind this? I mean, in Riemannian geometry, we always get a coordinate system defined by the charts(for simplicity I only use one) $\phi : M \rightarrow \mathbb{R}^k $ which we express here by $x_i := […]

Covariant derivative in $\mathbb{R}^n$

I am studying my lecture notes on covariant derivative, and is having difficulty to do a computation: Suppose $X,Y$ be smooth vector fields in $\mathbb{R}^n.$ Consider the integral curve $c_p:(-\epsilon,\epsilon) \to \mathbb{R}^n$ of $X$ passing through $p\in \mathbb{R}^n$ at time $t=0$. We have an identification $T_{c_p(t)}\mathbb{R}^n \to T_p\mathbb{R}^n$ of the tangent spaces. Using this, we […]

Geodesics of the $\mathbb{S}^n$ are great circles

I am trying to show that the geodesics of $\mathbb{S}^n$ are the great circles, as an exercise for my introductory Riemannian geometry class. I don’t really know how to go about this. I suppose that using the geodesic equation would be too complicated so I am trying to use the fact that if $M$ is […]