# Tagged Questions

A branch of differential geometry dealing with Riemannian manifolds. *Riemannian manifolds* are smooth manifolds with an inner product smoothly attached to the tangent space of each point. Usually, Riemannian geometry focuses on the notions of distance, curvature, and shape. Consider using this tag ...

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### Convolution of functions defined on manifold

Let $M$ be a Riemannian manifold with fixed volume form $\mu$. How to define a convolution of two 'functions' $f,g \in L^1(M)$? I will be grateful for an answer or for giving me some refrence where it ...
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### Measures which are absolutely continuous with respect to a Riemannian measure

Suppose $(M,g)$ is a oriented connected Riemannian manifold (but not necessarily compact). Let $\omega_g$ denote the volume form on $M$ determined by $g$, and let $m_g$ denote the probability measure ...
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### How to make sure any two points with small enough distance are inside a common open set

Let $K$ be a compact subset of a metric space, and I cover $K$ with finite open sets. How can I select $\delta>0$, such that for any $x,y\in K$, with $d(x,y) <\delta$, $x,y$ are inside an open ...
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### Role of Group actions in Differential Geometry

This is a rather soft question, my hope is to bring some order into the stuff I would like to learn about differential geometry -- here it is: I was told over and over again that Geometry has to do ...
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### Is there a charaterization of riemannian product manifolds?

Some Riemannian manifolds are expressed as a product manifold. Recently, I have read two articles about space-times. In both articles, the authors prove that a Riemannian manifold $\bar{M}^n$ is ...
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### Riemannian Metric of Lobatchchevski Geometry

I am stuck at a problem from Riemannain Geometry, written by Do Carmo. A function $g:\mathbb R\to\mathbb R$ given by $g(t)=yt+x$, $t$,$x$,$y\in\mathbb R$, $y>0$, is called a proper affine ...
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### Connections and Ricci identity

Given $\nabla$ a torsionless connection, the Ricci identity for co-vectors is $$\nabla_a \nabla_b \lambda_c - \nabla_b \nabla_a \lambda_c = -R^d_{\,\,cab}\lambda_d.$$ Prove $R^a_{[bcd]} = 0$ by ...
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### Global Bi-harmonic functions in a Riemannian manifold

Any help will be appreciated thanks! Consider $(\mathbb{R^n},g)$ to be a Riemannian manifold. For simplicity we can assume the manifold to be asymptotically euclidean outside a compact domain ...
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### Zero Sectional Curvature implies exp is a local isometry

Im studying DoCarmo's book Riemannian Geometry, the first problem of the chapter 5 (Jacobi Fields) states that If $(M,g)$ is a riemannian manifold with sectional curvature identically zero, show that ...
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### Alternative way to recognize that a real symmetric quadratic form is positive

A real symmetric quadratic form $g(x)=\varSigma_{1\leq i,j \leq n}\,g_{ij} x^i x^j$ is positive (definite) if $g(x)>0$ for every $\mathbb{R}^n\ni x=(x^1,...,x^n)\neq 0$. It is well known that a ...
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### Dimension of scalar solutions to these self-dual/anti-self-dual equations

Let $M$ be a 4-dimensional Riemannian manifold. Let $\kappa$ be a 1-form. I look for solution function $\phi$, such that there exists functions $\alpha$ and $\beta$ {\left( ...
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### Showing two forms on a manifold are equal

Let $\alpha$ and $\beta$ be two forms on a manifold $M$. To show that they are equal, does it suffice to show that for arbitrary $p\in M$ there exists some chart such that $\alpha_p=\beta_p$. I was ...
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### How small can an external angle of a circumference be if made of tangents?

Lets imagine the angle ABC where the lines AB and CB are tangents to a circumference which center is C. Lets assume that the points where the line AB touches the circumference is P and the point where ...
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### Prove the local expression of Riemannian curvature tensor

I try to prove the following expression of Riemannian curvature tensor: For local coordinate $\{x^i\}$, let $g_{ij}=g(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j})$ and ...
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### Maximum principle of harmonic function on compact manifold

Thm . (Maximum Principle) Let h be a harmonic function on a domain D in C . (a) If h attains a local maximum in D then h is constant. (b) Suppose that D is bounded and h extends continuously to the ...
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### Riemannian distance induced by an elliptic differential operator?

consider a Riemannian manifold $(M,g)$ and consider a second order elliptic differential operator. I've read that each such operator induces a riemannian distance function. Unfortunately I couldn't ...
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### If we don't need a Riemannian metric to compare length of vectors, why do we use metrics to measure curvature?

I read that, in the absence of a Riemannian metric tensor field, we can still measure how much a vector changes when parallel transported around a curve by comparing the initial and final vectors. ...
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### harmonic function on manifold

Let M be a 2 dimensional manifold. $h:M\rightarrow R$ be a harmonic function from manifold to real line. G is group that act by isometry. $g*h(x)=h(g(x))$. Let $W=\{x|h(x)=t\}$ that is the level set ...
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### Recommendable books to study the Selberg zeta function.

I've study on the Riemann zeta function and some zeta functions which have analytic properties directly. And now I want to know about the Selberg's zeta function which has some geometric properties. ...
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### How are “scalar curvature” and “sectional curvature” related?

I was browsing wikipedia and was puzzeling about what is the difference between: "scalar curvature" https://en.wikipedia.org/wiki/Scalar_curvature and "sectional curvature" ...
For some fixed $r_0>0,$ put the semi-Riemannian metric $$ds^2=\frac{r_0-r}{r}dt^2+\frac{r}{r-r_0}dr^2$$ on $\{(t,r)\in\mathbb{R}^2:r>r_0\}.$ I would like to show that the $r$-lines are always ...