Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange
Join us in building a kind, collaborative learning community via our updated Code of Conduct.

Questions tagged [riemannian-geometry]

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 if ...

0
votes
0answers
4 views

Volume form is parallel with respect to Levi-Civita connection

Let $(M,g)$ be a Riemannian manifold of dimension $m$. I would like to prove the following stament. The riemannian volume form $\omega$ is parallel with respect to Levi-Civita connection. This is my ...
0
votes
1answer
31 views

$\operatorname{grad}(f)$ definition and extra basis term.

Some preliminary definitions. On page 342 of Lee's Smooth Manifold he concludes that $\hat{g}(\operatorname{grad} f)(X) = Xf$ where $\hat{g}$ is the isomorphism between $TM \to T^*M$, the tangent ...
2
votes
0answers
25 views

Intuition behind discretization of first and second fundamental form in discrete differential geometry

I'm reading this paper with the hope I understand how to discretize the first and second fundamental forms. I think the intuition behind the discretization of the first fundamental form comes from the ...
0
votes
0answers
12 views

Flat tori as a riemannian product

A flat torus is defined as the torus with the metric inherited from its representation as the quotient $\mathbb{R}^2/\Lambda$ where $\Lambda$ is a discrete subgroup of $\mathbb{R}^2$ which is ...
1
vote
1answer
36 views

Some questions about Hermitian metric on complex manifold

Suppose $g$ is a Riemannian metric on a complex manifold $X$ compatible with the almost complex structure. $g$ can be extended to $TX\otimes\mathbb{C}$, which is a Hermitian inner product: $<\...
3
votes
0answers
35 views

Elliptic regularity of harmonic forms in $L^1$

$\newcommand{\M}{M}$ Let $\M$ be a smooth oriented Riemannian manifold. Let $\sigma$ be a differential $k$-form on $\M$ with coefficients in $L^1(\M)$. We say $\sigma$ is weakly harmonic if $$ \...
6
votes
1answer
57 views

A Hodge dual computation on a $4$-dimensional Riemannian manifold

Let $(M,g)$ be a $4$-dimensional smooth Riemannian manifold. I am trying to understand the following exterior algebra computation: Let $x^1,x^2,x^3,x^4$ be local coordinates on $M$ such that the ...
2
votes
0answers
22 views

Computing the geodesic curvature of curves in $\mathbb{H}^1 \times \mathbb{R} \subset \mathbb{L}^{3}$

Here $\mathbb{L}^3$ is $\mathbb{R}^3$ with the metric $\langle u, v\rangle_\mathbb{L} = -u_1v_1 + u_2v_2 + u_3v_3$, where $u = (u_1, u_2, u_3)$ and $v = (v_1, v_2, v_3)$ and $\mathbb{H}^1 \times \...
1
vote
1answer
25 views

Space quocient of Matrix Set

I'm studying some Hyperbolic hobby space, and I've been studying Geometry for some time now. Could someone explain to me what is homogeneous space and how should I fill this quotient?$\mathbf{H}^{3}\...
1
vote
0answers
32 views

Minimizing an integral involving first fundamental form

Suppose we have a surface parametrized by $\sigma$, (unknown) but suppose we also have parametrized surface $\sigma_0$ known I'm aiming to minimizing an integral involving a term like $$ F_I = \begin{...
0
votes
0answers
17 views

Orientability of hypersurfaces [on hold]

I often see that when the ambient manifold is simply connected, then any closed embedded hypersurface is automatically two sided and orientable, why is that? is it a simple fact?
1
vote
1answer
28 views

If some of the vectors in a dual basis are orthogonal then so are the original vectors?

Let $(V,g)$ be a $2n$-dimensional real inner product space. Let $v_i$ be a basis for $V$, and let $\theta^i$ be its corresponding dual basis. ($1 \le i \le 2n$). The metric $g$ induces a metric $g^*$ ...
0
votes
1answer
35 views

Why is $g(JZ,JW)=-g(Z,W)$?

Let $M$ be a complex manifold, with Riemannian metric $g$ and complex structure $J$. If $g$ satisfies $$g(JX, JY ) = g(X, Y ),$$ for any two vector fields $X$ and $Y$. I am reading a proof from here, ...
0
votes
0answers
42 views

Codifferential of volume form on Riemannian manifold.

When I read some lecture note about forms on Riemannian manifold, there was a formula $$\delta(fw_g)=-i_{\nabla f}w_g$$ Where $f\in C^{\infty}(M^n)$, $w_g$ is the volume form of $M$. $\delta$ is the ...
0
votes
1answer
24 views

Problem at proof of Cartan's theorem about the relation between metric and curvature in do Carmo's book

I'm reading DoCarmo's book, Riemannian Geometry and i don't understand something. At page 157, Cartan's theorem. My question is, why can we take a Jacobi field $J$ in such a way that $J(0)=0$ and $J(...
0
votes
0answers
22 views

inverse of a riem. metric in sobolev class still sobolev?

Given a covariant riemannian metric of certain sobolev class (i.e. with square-integrable weak derivatives up to a large enough integer order) on a (compact, if necessary) finite-dimensional smooth ...
1
vote
0answers
27 views

Question in the appendixes A of O'Neill book

I'm reading appendixes A of O'Neill's book, "Semi-riemannian geometry" and I don't understand a something. I don't understand at the last theorem, how we construct a universal cover for $M$ and why ...
2
votes
1answer
63 views

Can every smooth scalar function $f$ on $M$ be written as $f=\operatorname{div}(X)$?

Let $(M,g)$ be a Riemannian manifold of dimension $n$ and $X$ a vector field over it. divergence of $X$ is a real value function defined by: $$\operatorname{div}(X)=g(\nabla_{e_i}X,e_i),$$ where $\{...
0
votes
1answer
33 views

Hadamard theorem proof problem at the first lemma in do Carmo's book

I'm reading DoCarmo's book, Riemannian Geometry and i don't understand something. At page 149 first lemma I don't understand why at the end he says that "It follows that for all $t>0$..."
0
votes
0answers
31 views

Hopf-Rinow theorem from do Carmo's book, proof problem.

I'm reading do Carmo's book Riemannian geometry and I have a problem. At the proof of Hopf-Rinow theorem, more precisely at c.)=>d.) ($M$ complete metric space => $M$ geodezically complete space). I ...
0
votes
0answers
15 views

An inequality of p forms on Riemannian manifold

Suppose $(M^n,g)$ is an Riemannian manifold with boundary, $w$ is any $p$-form in $A^p(M)$, then we have following inequality $$\|\nabla w\|^2\geq \frac{\|d w\|^2}{p+1}+\frac{\|\delta w\|^2}{n-p+2}$$ ...
1
vote
0answers
14 views

How does one call/characterise sewn-together Riemannian manifolds

Suppose $(M_1,g_1)$ and $(M_2,g_2)$ are (semi)Riemannian manifolds with boundaries (resp.) $\Sigma_1$ and $\Sigma_2$. The metrics $g_{1,2}$ yield induced metrics $\gamma_{1,2}$ on those boundaries $\...
7
votes
0answers
53 views

Rigorous version of “spatial infinity is a point and not a sphere”

This is a question on differential geometry/topology of Lorentzian manifolds. It is motivated by Physics, but since I want a mathematicaly rigorous formulation I think the correct place to ask is here....
-2
votes
0answers
20 views

focal point (definition)

I am a begginer in the differential geometry and I need the definition of focal points. All the books I see is defined in riemannian submanifolds with jacobi fields. I know just the notions of ...
1
vote
0answers
15 views

Minimal surface contained in a hemisphere

I am looking for examples of closed, orientable minimal surfaces of the sphere $\mathbb{S}^3$ that are contained in a hemisphere. Do you know any? Thanks!
1
vote
2answers
22 views

total space of the bundle of orthonormal frames over a compact manifold is compact — how to see it?

I have seen this statement on several occasions but could not seem to be able to figure it out by myself. Surely the fibers are all compact but how does the compactness of the base (riemannian) ...
0
votes
0answers
21 views

Trace-free part of the second fundamental form

I found this notion on this part of a survey by S. Brendle: What exactly is the definition (without using basis) of the “trace-free part of the second fundamental form”? For me, the second ...
1
vote
2answers
45 views

Relation between gradient of a function with respect two linear connection

Let $\bar{\nabla}$ and $\nabla$ be two linear connection over $(M,\bar{g})$ and $(M,g)$ resp. and suppose that this two connection are related by the following equality: $$\bar{\nabla}_XY=\nabla_XY+\...
0
votes
1answer
46 views

Conjugate locus and local diffeomorphism problem in do Carmo's book.

I'm reading DoCarmo's book, Riemannian Geometry and i don't understand something. At page 149 first lemma My question is why $\text{exp}_p$ is local diffeomorphism?
2
votes
1answer
55 views

About the covariant derivative - is this correct?

Let $S \subset \mathbb{R}^3$ be a regular surface, the image of a parametrization $X: U \subset \mathbb{R}^2 \to S$ and $\alpha:I \to W \subset S$ be a curve in $S$. We can write $\alpha(t) = X(u(t), ...
1
vote
0answers
33 views

Diffusion maps & preserving the local geometry

I am reading about diffusion maps. It is stated that: the diffusion map preserves the local geometry of the graph." We know that this embedding is based on the Markov matrix. Now, I am curious to know:...
1
vote
0answers
24 views

Torus with sectional curvature zero

Ex. 2, Chap. 6 in do Carmo's Riemannian Geometry says $\mathrm{T}^2=\mathbf{x}(\mathbb{R}^2)\subset \mathrm{S}^3\subset \mathbb{R}^4$ is a torus with sectional curvature zero in the induced metric, ...
2
votes
0answers
37 views

Proof of Huisken's monotonicity formula

Firstly, I would like to say that I'm studying by myself Riemannian Geometry in order to be able to understand Mean Curvature Flow, so there are some computations that I don't understand well (as I ...
3
votes
1answer
42 views

Systematic application of algebraic topology to energy minimization problems?

I have come across two different occurrences of energy minimization problems which find an interpretation using notions from algebraic topology, and I was wondering whether analogous situations have ...
2
votes
0answers
20 views

Riemannian metric on an Oriented Riemannian manifold $X$ induces an Hermitian inner product

The Riemannian metric on an Oriented Riemannian manifold $X$ induces an Hermitian inner product $<,>$ on $\wedge^pT^*_xX$, I wonder how is the Hermitian inner product defined?
1
vote
0answers
34 views

Isometric immersion problem in do Carmo's book

I'm reading do Carmo's book, Riemannian geometry and I don't understand some things at the page 126. I know that on a Riemannian manifold, $(\nabla_X Y)_p$ depends on $X^i_p,Y^i_p$ and $(\partial_iY^...
0
votes
1answer
33 views

Two approaches to the definition of a connection

A connection can be defined as a map $$D:\Gamma(E)\to\Gamma(E)\otimes\Gamma(T^*M)$$ satisfying the usual conditions or, equivalently, $$F:\Gamma(TM)\otimes \Gamma(E)\to \Gamma(E).$$ My question is ...
2
votes
2answers
69 views

How are Gaussian curvature and Riemann curvature tensor related?

Is the Riemann curvature tensor generalization of the Gaussian curvature?
2
votes
1answer
97 views

Finding local orthonormal frame on a Pseudo-Riemannian Manifold

Suppose we have a semi-Riemannian manifold $(M^n,g)$ with metric signature $(n-k,k)$. By definition, each $p \in M$ the map $g_p : T_pM \times T_pM \to \Bbb{R}$ is a non-degenerate, symmetric, ...
0
votes
0answers
44 views

What is the complexification of the $n$-sphere $S^n$?

What is the complexification of the $n$-sphere $S^n$ under the construction given in this post? Does such a construction exist? I do not think it would be $\mathbb{CP}^{n/2}$, as this is not a ...
1
vote
1answer
64 views

Understanding the definition of norm of tensors on a Riemannian manifold

I am teaching myself Riemannian Geometry in order to studying Mean Curvature flow. I was reading Lecture Notes on Mean Curvature Flow by Carlo Mantegazza and I'm trying understand the following ...
4
votes
0answers
77 views

Mistake in do Carmo's book Riemannian geometry?

I'm reading do Carmo's book Riemannian geometry and at the chapter 5 Jacobi fields, is this proposition. My question is where is $a$ in the proof? At the first equal sign we should have $\frac{1}{a}$...
1
vote
0answers
21 views

Constructing metric tensor of cosmological models viewed as (Riemann) homogeneous space and connections with the Killing vectors

As we know cosmological models are Riemann manifolds which are assumed to have some sort of symmetries (spherical, isotropic, homogeneity and etc.) and the problem is to find the form of the metric ...
2
votes
1answer
30 views

How can I obtain a normal unitary field to higher dimensional surfaces?

More specifically, say I've got a surface in the form $X(u,v) = (x(u,v), y(u,v),z(u,v),t(u,v))$. The tangent fields are obviously $X_u$ and $X_v$, but I don't know how to give meaning to $X_u \times ...
3
votes
1answer
55 views

How bad can the intersection of two totally geodesic submanifolds be?

Let $M$ be a complete Riemannian manifold and let $S_1,S_2 \subset M$ be totally geodesic submanifolds which are closed as subspaces of $M$. Question: Is $S_1 \cap S_2$ a submanifold of $M$? Or is it ...
1
vote
0answers
84 views

Morse–Palais Lemma and local approximation

Let $M$ be an $n$-dimensional Riemannian manifold and let $E:\Lambda M\to\mathbb{R}$ be the energy functional $E[\gamma]=\int_{S^1}||\dot\gamma(t)||^2dt$ where $\Lambda M$ is the free loop space of $M$...
3
votes
1answer
62 views

How can one generalize the Gauss map to higher dimensions? More specifically, bi-dimensional manifolds in $\mathbb{R}^4$

It's easy to define the unitary tangent fields of a $2$-dimensional surface $S: I \times J \to \mathbb{R}^4$, but since I don't have the cross product anymore, an unitary normal field is harder to ...
1
vote
0answers
22 views

Why is a connection on the bundle $SO(M)$ metric compatible?

If we have an orientable manifold $M$ with a metric $g$ and signature $(r, s)$, we can define the principal-$SO(r, s)$ bundle $SO(M)$, the bundle of orthonormal frames of $TM$. This is a subset of the ...
0
votes
0answers
31 views

Exponential map on Heisenberg group is a diffeomorphism.

Assume that a Riemannian manifold $M$ is simply connected. In further, assume that there is an global orthnormal frame $e_i$. If $f_i$ is flow of $e_i$, and $M$ is homeomorphic to $\mathbb{R}^n$, then ...
2
votes
2answers
103 views

ODE problem reading do Carmo's book of Riemannian geometry

I'm reading do Carmo's book, Riemannian geometry. I have a problem at the Jacobi fields. He talks about the case of constant curvature. He gets to the ODE above and my problem is how dose he solve it? ...