# Tagged Questions

48 views

### Are the topology of a manifold and the topology induced by the metric of a manifold the same?

I am a physics student trying to understand in a rigorous way what a manifold is so please bear with me. Ok, so I am just learning what a topology is and from what I have understood up till now is ...
66 views

### Is there any difference between a flat manifold and an affine space?

What is the difference, if any, between a flat manifold (in which the Riemann tensor vanished identically) and an affine space?
28 views

### Example of complete not-connected riemannian manifold

Are there examples of complete Riemannian manifolds which are not connected ? This question follows my previous question. The more I think about it and the less I'm convinced it exists.
70 views

### Riemannian metric and geodesic

For all $p \in \mathbb{R} \smallsetminus \lbrace 0,1 \rbrace$, let $\displaystyle M(p)=\frac{1}{p^{2}(p-1)^{2}}$. Then, let $g_{p} \, : \, (u,v) \, \longmapsto \, uM(p)v$. I am not sure about the ...
31 views

### Difference betwee parameterization and embedding of manifolds

What is the difference between embedding and parameterization? Why, for example, we say Gauss parameterization of a convex hypersurfaces, and we don't call it an embedding?
46 views

### extension of a local orthonormal frame on a hypersurface

Let $N$ be a $(n+1)$-dimensional Riemannian manifold and $M\subset N$ a Riemannian hypersurface (embedded or immersed). Let $M$ and $N$ be oriented and choose a unit normal vector field $\nu$ along ...
35 views

### Question about a particular estimate in Riemannian geometry.

I have been studying the book Some Nonlinear Problems In Riemannian Geometry - Thierry Aubin. On page $46$ he begins the proof of the Sobolev imbedding theorem to manifolds. The proof is divided in ...
15 views

### Why $\dim(K\cap N(I)) +1 \geq \dim(K)$ if the index of the index form equals 1?

Let $c:[0,1]\rightarrow O$ a geodesic, $O$ Riemann manifold and let $\mathcal{W}$ the space of piecewise smoothl normal vector fields $W(t)$ along $c$ mit $W(0)=W(1)=0$. $N(I)$ is the nullspace of ...
62 views

### Calculate Geodesic Path of $N\times N$ matrix on Riemannian manifold of fixed rank

If I have two matrices $A(0)$ at $t=0$ and $A(1)$ at $t=1$, they are $N\times N$ matrices, and they are on the Riemannian manifold of rank $K$. How to calculate the geodesic path $A(t)$? I haven't ...
53 views

### Decomposition of Laplacian into tangental and normal components w.r.t. submanifold

If I have the covariant Laplacian operator acting on a tensor e.g. $\nabla^2 h_{\mu\nu}$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) submanifold, how can I "decompose" the ...
46 views

### Calculate geodesic path on matrix manifold

I have a matrix which is change with time. Let me denote it as A(t). I know t=0 it is A(0) and I know t=1 it is A(1). A is symmetric positive semi-definite matrix. What I want to do is find the ...
20 views

### Boundedness of Riemannian curvature gradient

I'm reading a paper by Wan-Xiong Shi "Deforming the metric on complete Riemannian manifolds". And there is a statement without proof. It can be summarized as follows: Let $B(x_{0},\gamma)$ be a ...
20 views

### Reformating Function

Is there such a function where a ambiguous ;n-dimensional, field/space (defined by a function) is plugged in and returns a flattened field where the basic units along the function are then formatted ...
39 views

### Almost complex structure compatible with Levi-Civita connection of immersed submanifold?

Suppose we have Riemannian manifold $M$ isometrically immersed in a Kähler manifold $\tilde{M}$. Let $\tilde{g}$ be the metric and $\tilde{\nabla}$ the Levi-Civita connection on $\tilde{M}$ , we know ...
60 views

### Which of principal curvature, Gaussian curvature and mean curvature is intrinsic ？ Why?

Which of principal curvature, Gaussian curvature and mean curvature is intrinsic and why? Which book will have this explanation? I read a book (Remannian Geometry, Do Carmo, p129). It has a saying: ...
49 views

### When can you recover a connection from totally geodesic submanifolds?

Let $g_{ab}$ a Riemaniann ( Lorentzian ) metric in a $n-$dimensional manifold $N$ and let $M$ be a submanifold of $N$. In general, the Levi-Civitta connection induced by the induced metric in $M$ ...
15 views

### The time evolution of levi-Civita connection

Assume a smooth one-parameter family of Riemannian metrics $g_{t}$. Write $h:=\frac {\partial}{\partial t}g$. In addition, assume that the Levi-civita connection on the Riemannian manifold ...
209 views

### Prerequisite for Petersen's Riemannian Geometry

A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. ...
60 views

### Definition of the integral of a vector field on Riemannian manifold and Euclidean spaces

Given a compact Riemannian manifold $(M,g)$ and a vector field $X \in \mathfrak{X}(M)$, is it possible to define the integral of $X$ on $M$? What if $M$ is a Euclidean space? Clearly the definition ...
64 views

### Completeness of a Riemannian manifold with boundary

I have some issues understanding the notion of completeness of a Riemannian manifold with boundary. In the case of Riemannian manifolds without boundary, I found that completeness is usually defined ...
87 views

### Is a bounded (Riemannian) manifold always totally bounded?

Let $(M,g)$ be a finite dimensional Riemannian manifold. Suppose it is bounded as a metric space, i.e. $$\sup \{ d(x,y) | x,y \in M \} < \infty$$ where $d$ is the distance induced by $g$. Is $M$ ...
82 views

### Calculus on manifolds

I am studying manifold theory ( smooth manifold). As I study the subject, I feel more and more confused. I am not yet at chapter dealing with deferential forms and integration. I can understand that ...
62 views

### measure on non-oriented Riemannian manifold

Let $M$ be a non-oriented Riemannian manifold of dimension $m$. Nash embedding theorem implies that there exists an isometric embedding $\phi: M\longrightarrow \mathbb{R}^n$ for $n$ sufficiently ...
87 views

### Notation and hierarchy of cartesian spaces, euclidean spaces, riemannian spaces and manifolds

I am confused by some definitions. Forgive the looseness of my language. A Cartesian space is basically a space of points that can be represented by n-tuples ( and other things, but I won't go into ...
76 views

### Is this a geodesic?

Let $(M,g)$ be a riemannian manifold. Let $p$ in $M$ and $v,v_{0}$ two vectors in $\mathrm{T}_{p}M$. I am looking at the curve $$\gamma \, : \, t \, \longmapsto \, \mathrm{Exp}_{p}(tv+v_{0})$$ ...
74 views

### Supremum Definition in context of a Rectifiable/Differentiable Curve

I am trying to prove that if I have 2 paramterizations of the same curve $\gamma$ and $\sigma$ (i.e. there is continuous bijective map $\phi$ such that $\sigma = \gamma \circ \phi$) then if the curve ...
163 views

### What does it mean “being geodesic” is not invariant?

We know that "being geodesic" is not invariant under re-parametrization. Only affine re-parametrization preserves the property of being a geodesic. Also, a geodesic is locally distance minimizer. ...
104 views

### Geodesics of one-dimensional manifold

I apologize if my post is "silly" because I don't know much about riemannian geometry. I know that $M = (0,1)$ (the open unit interval) can be seen as a one-dimensional manifold. Since $M$ is an ...
55 views

### Finding the components of the Riemannian tensor given the components of a metric.

I am looking at a manifold of dimension $n$ (And I am considering a local co-ordinates system $x_1,x_2,\ldots x_n$) and the metric defined by the components $g_{ij} = \frac{\delta_{ij}}{x_1^2}$. I'm ...
61 views

### Elliptic Regularity on Manifolds

Sorry if this has been asked before. Are there are books which talk about elliptic regularity on manifolds? Everything I've been able to find has been about $\mathbb{R}^n$. Can every question about ...
291 views

### Another vector bundles over of a Riemannian manifold

How is that any other vector bundle over a Riemannian manifold can be turn into a Riemannian manifold as well? I think that the question is of interest due to the presence in math.SE of several ...
102 views

### Submanifold is complete

If $M$ is a complete manifold and $N\subset M$ is a closed, embedded submanifold with the induced Riemannian metric, show that $N$ is complete. I really don't know where to start. This is not ...
57 views

### Metric tensors. Have I got the correct understanding?

My course is covering metric tensors in a slap-dash way, so I want to ensure I have understood correctly how they are described. I hope you can confirm this! So, I believe that a metric tensor is a ...
75 views

### Why is this curve a topological manifold?

Why is $$M=\{(z_1,z_2)\in \mathbb{C}^2 \, |\,\, z_1^3-z_2^4=0 \}$$ a topological manifold? I understand for example why why $|z|=1$ is a topological manifold, since I can write every point as ...
72 views

39 views

### Explanation required of the following definition:

This is a definition I encountered in a paper. I hope someone will be able to help me understand it. The authors assume a Frenet curve $\alpha(s)$ on a 3-D Riemannian Manifold as any non-geodesic unit ...
298 views

### Lie bracket is a connection?

In Road to Reality, section 14.6 on Lie derivative Penrose writes: Now $\epsilon^2 [j,h]$ corresponds to an $O(\epsilon^2)$ gap in the ‘parallelogram’ whose initial sides are $e_j$ and $e_h$ at ...
80 views

Given: two Euclidean spaces $\mathcal{P},\mathcal{P'}$ (with their usual smooth structure) and a injective smooth mapping $f:\mathcal{P}\to\mathcal{P'}$ a Riemannian manifold $(\mathcal{M}\subset ... 0answers 56 views ### warped products Problem: Consider the following warped product$M^{n+1}=\mathbb{R}\times_{f} \mathbb{P}^{n}$, where$\mathbb{P}$is a complete n-dimensional Riemannian manifold,$f:\mathbb{R}\rightarrow\mathbb ...
Could you please provide me with the precise definition of isotropic curves of a conformal structure on a manifold $M$? If there is such a definition, then can I say the following: if $c$ is an ...