# Questions tagged [riemannian-geometry]

For questions about Riemann geometry, which is a branch of differential geometry dealing with Riemannian manifolds.

5,990 questions
Filter by
Sorted by
Tagged with
13 views

### Proof that $D_XY=(X^i\partial_i Y^j) \partial_j$ is metric connection on $\mathbb{R}^3$

I need to proof $D_XY=(X^i\partial_i Y^j) \partial_j$ is metric connection on $\mathbb{R}^3$ I want to use metric tensor (dot product). So I must show that $Z(X \cdot Y)=D_ZX \cdot Y +X \cdot D_ZY$. I ...
10 views

### Surface gradient of a vector field

Let $\mathbb{M} \subset \mathbb{R}^3$ be a $C^2$ manifold with normal vector ${\bf n}$. Then, for any scalar-valued function $f$, we can write its surface gradient as: \begin{align} \nabla_{\mathbb{M}}...
34 views

### When can the trajectories of a vector field be realized as geodesics?

Suppose we have a smooth manifold $M$ and a vector field $X$ on $M$. Then the trajectories of $X$ are pairwise non-intersecting, and further, each trajectory is either injective or periodic. I'm ...
39 views

### The Ricci tensor is independent of the coordinates.

Could you help me with the following problem, please? What I will write can be found in the book Introduction to Riemannian geometry with applications to mechanics and relativity by Leonor Godinho and ...
25 views

### Changing the Riemannian metric does not affect the $Spin^c$-structure

It is well known that changing the Riemannian metric on a manifold does not change the $Spin$-structure. I suppose the same should be true for $Spin^c$-structures, but I am unable to prove it or to ...
57 views

### Spectrum of $\Delta+a$ on a compact manifold, where $a$ is a function

Let $(M,g)$ be a closed Riemannian manifold and $\Delta$ be its Laplace-Beltrami operator. It is well known that the spectrum of $\Delta$ is a discrete subset of $[0,+\infty)$ and the eigenfunctions ...
28 views

24 views

### Regular embedded manifold, isometric embedded

In "On Einsteins Path" Chapter 11 "Wave Maps in General Relativity", I have seen the following definition for a "regularly embedded manifold $(M,g)$: Let $q$ be a Euclidean ...
46 views

### The hyperbolic plane $\mathbb{H}^2$ can't be isometrically immersed in $\mathbb{R}^3$

It's easy to note that there is a local isometry between the hyperbolic plane $\mathbb{H}^2$ and the pseudosphere, since they have constant curvature equal to $-1$. Hilbert's Theorem.- There exists no ...
46 views

### Example of Riemannian metric on sphere such that it become non strictly convex.

My understanding of strictly convexity of the compact set in Euclidean space is that if we take any straight line joining any two boundary points then the line must be in a compact set with out ...
30 views

### Mean curvature equals to the Laplacian of the position vector?

My question is based on the manuscript titled On Mean Curvature Diffusion in Nonlinear Image Filtering (link: https://escholarship.org/content/qt0736r63m/...
51 views

### What is the analogous of $\|\nabla X\|^2=\sum_{i=1}^ng(\nabla_i X,\nabla_i X)$ for two forms?

For a vector field $X$ it is well-known that $\|\nabla X\|^2=\langle\nabla X,\nabla X\rangle=\sum_{i=1}^ng(\nabla_{e_i} X,\nabla_{e_i} X)$ where $\{e_i\}$ is an ONB and $\langle \cdot,\cdot \rangle$ ...
54 views

### Regularly embedded manifold

In "On Einsteins Path" Chapter 11 "Wave Maps in General Relativity", I have seen the following definition for a "regularly embedded manifold $(M,g)$: Let $q$ be a Euclidean ...
33 views

### Compact manifold with smooth boundary embedded isometrically into compact manifold without boundary with same dimension as original one.

Let $(M, g)$ be a compact manifold with a smooth boundary. There is a compact connected manifold without boundary $(N, g)$ having the same dimension as $M$ so that $(M, g)$ is isometrically embedded ...
36 views

55 views

40 views

### Globally hyperbolic Lorentzian manifold

I am currently trying to work through properties of globally hyperbolic Lorentzian manifolds and there are some things which aren't clear to me: I have the following definition of globally hyperbolic: ...
58 views

### Extrinsic definition of a Riemannian manifold?

In every book that I have read (Bott and Tu, Absil), Manifolds are defined intrinsically in terms of charts and maximal atlases, after which we show how to "construct" manifolds from level ...
45 views

### Finite volume conditions of manifolds

I was reading this question and wondering if there are more general results answering the question "Under which condition do manifolds have to have a finite riemannian volume?". I know that ...
35 views

### Geodesic orthocenter

Suppose $(M,g)$ is a complete Riemann manifold. Is it true that for any $A,B,C \in M$ that do not lie on the same geodesic, there exists an $H \in M$ such that $A,B,C,H$ lie in a totally geodesic ...
39 views

### In what sense is the Ricci Curvature Tensor an “average”?

I’ve heard that the Ricci tensor loosely represents an “average” curvature of some sort. The physical meaning of the Riemann tensor is relatively clear to me as I’ve seen how to derive it, but the ...
### Relationship between the $\mathtt{Exp}^{-1}$ mapping of three points on a common geodesic on a Riemann manifold
Suppose that there is a Riemann manifold $\mathcal{M}$ and $x,z$ be two point on $\mathcal{M}$. Let \begin{equation} y = \mathtt{Exp}_{x}\left( t \cdot \mathtt{Exp}_{x}^{-1} \left( z \right)\right) \...