# Questions tagged [riemannian-geometry]

Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.

1,831
questions

**3**

votes

**0**answers

63 views

### Geodesics (Local vs Global)

Let $M$ be a Riemannian manifold, and $p,q\in M$ two points. Now, if $M$ is a complete metric space the Hopf–Rinow theorem ensures that there is a geodesic $C\subset M$ joining $p$ and $q$ that ...

**2**

votes

**0**answers

80 views

### Volume doubling, uniform Poincaré, counterexample

The Poincaré inequality and the volume doubling property are important notions related to heat kernel estimates.
Pavel Gyrya and Laurent Saloff-Coste obtain the two sided heat kernel estimate of ...

**4**

votes

**0**answers

99 views

### “Inflating” a closed, defined metric, manifold

Consider a two-dimensional Riemannian manifold homeomorphic to the sphere, with a defined metric.
Since we do not suppose that manifold to have a positive curvature,
we are not in the hypotheses of ...

**2**

votes

**0**answers

66 views

### Convergence of non-collapsing sequences of Riemannian manifolds with biliterally bounded sectional curvature

EDIT: I heard that there is the following result:
Given a sequence $\{(M_i^n,g_i)\}$ of compact smooth $n$-dimensional Riemannian manifolds with uniformly bounded absolute value of sectional ...

**12**

votes

**2**answers

396 views

### Possible isometry groups of open manifolds

Consider a non-compact manifold $M$.
Does there always exist a Riemannian metric on $M$ such that the isometry group is non-compact?

**2**

votes

**1**answer

70 views

### Gaussian null coordinates

I find it hard to find information on the so-called "Gaussian null coordinates", which Wikipedia says is used to describe "near horizon geometries". Can someone provide a reference where I can read ...

**-1**

votes

**0**answers

116 views

### Norm of a Riemannian metric

I'm writing a thesis and I need to be able to say when two Riemannian metrics are close. I was reading a paper and there the definition was assumed to be known, so I guessed it had to be something ...

**1**

vote

**1**answer

114 views

### Lipschitz constant of exponential map

I asked before this question on MSE but I was not able to work out the details on my own.
Suppose $M$ is a smooth compact Riemannian manifold, take $p \in M$ and consider the map
$$ T_pM \ni v \...

**1**

vote

**0**answers

72 views

### Gradient of squared riemannian distance on complete manifold

Let $\theta: M \times M \to \mathbb{R}$ the squared distance function $\theta(x,y)=d(x,y)^{2}$ on complete Riemannian manifold $M$. I would like to calcule the gradient of $d^{2}$, where $d^{2}_{y}(x)=...

**5**

votes

**0**answers

125 views

### Embed the hyperbolic plane into Euclidean spaces

Can the complete simply-connected surface with constant Gauss curvature -1 be embedded smoothly in the 5-dimensional Euclidean space?
Can the complete simply-connected surface with constant Gauss ...

**2**

votes

**1**answer

68 views

### Compact group actions with uniformly bounded derivatives

Suppose we have a smooth action of a compact Lie group $G$ on a non-compact smooth manifold $M$, denoted by
$$\phi:G\times M\rightarrow M.$$
Differentiating $\phi$ at a point $x\in M$ gives a map that ...

**2**

votes

**0**answers

81 views

### Volume form on unit normal bundle via moving frames

Let $M$ be an $m$-dimensional Riemannian submanifold of $\mathbb{R}^{m+n}$. Let $B_1$ denote the unit normal bundle of $M$, whose fiber at $p \in M$ is the $(n-1)$-sphere $\mathbb{S}^{n-1}$ in the the ...

**0**

votes

**0**answers

38 views

### On the Variable Coefficient Laplacian

This the copy of the question that I had asked in math stackexchange
I read about Laplace Operator here. As given in the link, given the metric, we can find the expression for Laplace operator. I am ...

**2**

votes

**0**answers

26 views

### Cylindrical coordinates in quotient of symmetric space

I am interested in the following situation. Suppose $G/K$ is a symmetric space of non-compact type and $\alpha$ is the axis of a hyperbolic isometry. I am interested in computing the Hessian of the ...

**7**

votes

**2**answers

194 views

### Is the injectivity radius (semi) continuous on a non-complete Riemannian manifolds?

Let $\mathcal{M}$ be a Riemannian manifold, and let $\mathrm{inj} \colon \cal M \to (0, \infty]$ be its injectivity radius function.
It is known that if $\cal M$ is connected and complete, then $\...