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 ...

learn more… | top users | synonyms

2
votes
0answers
41 views

subharmonic function and support functions

$M$ is a Riemannian manifold and $f$ is a continuous function on $M$. $f$ has the property that for any $p \in M,\epsilon>0$, we can find a smooth function $f_{\epsilon}$ such that ${f_\varepsilon ...
1
vote
0answers
43 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 ...
1
vote
2answers
519 views

An Einstein manifold has constant scalar curvature.

I know this is called Schur's lemma. But I cannot find a proof. All references available to me either does not give a proof, or says that it is similar to the lemma for sectional curvature, making use ...
1
vote
1answer
91 views

Lifting problems existence

Let $g:\mathbb{R}^m \longrightarrow \mathbb{R}^m,g\in C^1(\mathbb{R}^m)$ such that: $\|g'(x)(v)\|\geq\|v\|,\forall v\in \mathbb{R}^m,\forall x \in \mathbb{R}^m$ show that any rectilinear path ...
2
votes
0answers
131 views

Question from Peter Petersen

I'm trying do a exercise from Peter Petersen's book, but I don't know what do. Well, assume that $$R=\frac{scal}{2n(n-1)}g\circ g+\left(Ric-\frac{scal}{n}g\right)\circ g+W$$ Where, R is the ...
3
votes
0answers
170 views

Every holomorphic map between Kähler manifolds is harmonic

I was reading the Wikipedia article on harmonic maps and saw the following statement in the 'examples' section: Every holomorphic map between Kähler manifolds is harmonic. I am not that familiar ...
4
votes
0answers
83 views

Gauss-Bonnet Theorem in dimension four

I've read that the generalized Gauss-Bonnet theorem states that $$\int\limits_{M}Pf(\Omega)=(2\pi)^n\chi(M)$$ where, $M$ is a 2n-dimensional compact orientable Riemannian manifold without ...
4
votes
1answer
250 views

Need help finding a good book on Riemann Geometry

I want to learn more about calculus on manifolds and Riemann Geometry. I have been reading the book Geometry, Topology and Physics by Nakahara. But I find that it is difficult to read due to the lack ...
6
votes
2answers
574 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 ...
1
vote
1answer
372 views

Convert Euclidean distance to Hyperbolic distance

I am searching for formulae to convert Euclidean distance into hyperbolic distance. The problem that I am confronting is that I want to calculate if a bug travels $x$ units in Euclidean space, how ...
3
votes
1answer
374 views

How to convert between the hyperboloid model and the Poincare patch for $\mathbb{H}_n$?

This question is motivated by the results in this paper, http://calvino.polito.it/~camporesi/JMP94.pdf In this paper some of its most important results about the asymptotics of symmetric traceless ...
5
votes
1answer
133 views

Metric on Riemannian manifolds

Why is it necessary to consider taking the infimum over the lengths of all piece-wise smooth curves while defining the distance function on a Riemannian Manifold instead of just taking the infimum ...
2
votes
0answers
45 views

Diagonalization of Curvature Operator of $P^n({\bf C})$

Consider $P^n({\bf C})$ which is a quotient of $(S^{2n+1}, {\rm can})$. If $ \{e_1, ... , e_n, Je_1, ... , Je_n\}$ is a basis on $T_xP^n({\bf C})$ where $J$ is an almost complex structure, then ...
4
votes
0answers
75 views

bounds on eigenvalues of elliptic operators on functions on riemannian manifolds

Well I have little experience with pde's and analysis, I mostly study topic related to geometric topology and I would like to see if someone can please explain to me why is it important to find bounds ...
9
votes
1answer
257 views

Creating geodesics on manifolds

Suppose I have two points on a Riemannian manifold $M$, called $p_0$ and $p_1$. I have a family of curves $\gamma:[0,\infty)\times[0,L]\to M$ such that $\gamma(t,0) = p_0$ and $\gamma(t,L) = p_1$. ...
4
votes
1answer
97 views

Smooth mapping between Manifolds

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 ...
1
vote
1answer
89 views

Boundary Normal Coordinates

Let $ (M,g) $ be a 2D Riemannian manifold with boundary. The boundary normal coordinates $\psi $ are constructed roughly as follows: in a sufficiently small neighborhood $ U $ of $ \partial M $, for ...
2
votes
0answers
94 views

Euler Characteristic in dimension four

My doubt is simple: How can I prove that in dimension 4, the Euler Characteristic of a Riemannian Manifold is given by ...
1
vote
0answers
118 views

Does parallel Ricci tensor imply constant scalar curvature?

Is that true that if a Riemannian manifold $(M,g)$ has parallel Ricci tensor i. e., $\nabla Ric=0$, then the manifold has constant scalar curvature? I've seen this result with some aditional ...
1
vote
1answer
245 views

Decomposition of the Curvature operator and Matrix representation

I'm trying do this question from Peter Petersen's Book and I can't do some parts. I know that $$R=\frac{scal}{2n(n-1)}g\circ g+\left(Ric-\frac{scal}{n}g\right)\circ g+W$$ Where, $R$ is the ...
3
votes
1answer
84 views

Block Decomposition of a linear map on $\Lambda^2TM$

I'm trying a exercise from Peter Petersen's book, and I did the following: Let $*$ be the Hodge star operator, I know that $\Lambda^2TM$ decompose into $+1$ and $-1$ eigenspaces $\Lambda^+TM$ and ...
1
vote
1answer
68 views

Reflection along subspace

A symmetric space is a Riemannian manifold M with the following property: For every point $p \in M$ there is an isometry $\phi: M \rightarrow M$ such that $\phi(p) = p$ and $\phi_*(v_p) = -v_p \in ...
5
votes
1answer
172 views

The Ricci flow and $\frac{\partial}{\partial t}g_{ij}=-2(R_{ij}+\nabla_i \nabla_j f)$ are equivalent up to diffeomorphism

Suppose $M$ is a Riemannian manifold. Consider flow $\frac{\partial}{\partial t}g_{ij}=-2(R_{ij}+\nabla_i \nabla_j f)$, where $f$ is a time-dependent function. I would like to prove that flows of this ...
0
votes
1answer
64 views

Way distinguishing whether or not complex manifold

$SU(3)$ has dimension 8. Why is this not a complex manifold ? Thank you in advance.
1
vote
2answers
209 views

How can I compute the area of a geodesic triangle?

How can I compute the area of a geodesic triangle in a Riemannian 2-manifold? If the Gauss curvature $K$ is constant and positive I can take the Gauss-Bonnet theorem to obtain ...
2
votes
1answer
104 views

Confused by local isometries

I think I am a little confused about the notion of a local isometry of Riemannian manifolds. Let's say I have a manifold $(M,g)$ where $g$ is the Riemannian metric. Take a chart $x:U \rightarrow ...
2
votes
1answer
230 views

How to compute Bochner laplacian $\Delta=\nabla^*\nabla=\sum \nabla_{e_i}$?

I'm struggling with proving that Bochner laplacian can be described by the following formula similar to the standard laplacian formula from calculus: $$\Delta = \sum_i \nabla_i^2,$$ where $\nabla_i = ...
3
votes
0answers
81 views

Distance function and Green's function

Let $ (M,g) $ be a two dimensional Riemmanian manifold, with a smooth distance function $ d(x,y) $ for all $ x, y$ in $ M$. The logarithm of the distance function then satisfies $ \Delta \left( ...
2
votes
1answer
157 views

Self-dual and anti-self-dual decomposition

Please take a look at the following: Let $(M,g)$ be a four-dimensional oriented Riemannian manifold. The Hodge star operator $*$ obeys $**=Id$ acting on 2-forms. This allow us to decompose the ...
8
votes
1answer
252 views

Is $ds$ a differential form?

I am somewhat confused as to whether $ds$ (line element) is actually a differential form... we have (in $\mathbb{R}^2$): $$ds^2 = dx^2 + dy^2$$ Differential 1-forms are supposed to be linear ...
1
vote
0answers
71 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 ...
3
votes
1answer
170 views

Is there a Smooth Real Manifold which is not a Riemannian Manifold?

I am taking a course in Differential Topology right now, but I know of another Subject called "Riemannian Geometry" which studies Riemannian Manifolds. The definition of a real smooth manifold and a ...
6
votes
1answer
175 views

Geometric Interpretation: Parallel forms are harmonic

Let $(M,g)$ be a Riemannian manifold. The canonical volume form $\mu=\sqrt{\det g_{ij}}\mathrm{d}x^1\wedge\dots\wedge\mathrm{d}x^m$ is parallel w.r.t. the induced Levi-Civita conection $\nabla$ ...
2
votes
1answer
81 views

General formula to calculate the divergence of an operator

Le $M$ be a Riemannnina manifold. Is possible to express the divergence of an $(1,1)$ tensor $T$, i. e., an operator by a general formula as happen for the Ricci tensor where we have the well known ...
10
votes
2answers
315 views

Are closed geodesics the prime numbers of Riemannian manifolds?

I wonder to what extent one can support the analogy that primitive closed geodesics are the prime numbers of Riemannian manifolds? ("Primitive": traced once, as opposed to $m$-fold for $m \ge 2$.) In ...
0
votes
1answer
108 views

Inclusion mapping in conformal compactifications

The conformal compactification of the prototype Euclidean space $\mathbb R^n$ can be carried out by embedding $\mathbb R^n$ into $\mathbb R^{n+1,1}$ with Minkowski signature. One then considers the ...
2
votes
1answer
139 views

Precise definition of isotropic curve of a conformal structures on a manifold?

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 ...
4
votes
1answer
307 views

What is a principal orbit

I am currently reading a paper about Einstein manifolds. There is a comment where I don't know exactly the meaning of the words, namely a certain metric has a group of isometries of dimension $4$ ...
3
votes
1answer
66 views

Is a geodesic the least curved path?

It is clear that, in $\mathbb{R}^n$, straight lines are the lines with minimum possible curvature. That is, given the Frenet-Serret ($n$-dimensional equivalent) matrix, and taking its squared norm, ...
3
votes
1answer
121 views

Harmonic maps from the sphere

In Hamilton's 1997 paper on four-manifolds with positive isotropic curvature, he considers a local diffeomorphism of Riemannian $n$-manifolds $$ P: (N,\bar g) \to (M, g). $$ Such a map is harmonic if ...
8
votes
1answer
202 views

What kind of polygonal surface has an interior angle > 360°?

Consider this polygon as the setting for a dynamical billiard: When it's drawn in the plane, the polygon intersects itself; it is non-simple. However, I don't want to embed the polygon in the ...
2
votes
1answer
52 views

Defining $W^{k,p}(M)$ for non-integers $k$ and $p$ and manifold $M$

For $k$ and $p$ not necessarily integer, and on a smooth manifold $M$, how to define the Sobolev space $W^{k,p}(M)$? I've only seen definitions for $p=2$.
8
votes
2answers
320 views

Total mean curvature of an immersed torus.

How to prove that the total mean curvature of an immersed torus of $R^3$ such that has nontrivial self-intersection must $> 8 \pi$? The definition of total mean curvature is the integral of $H^2$ ...
3
votes
1answer
50 views

volume and curvature of submanifolds

Suppose an $m$-dimensional manifold in an $n$-dimensional euclidean space, choose some point on this manifold and take an $n$-dimensional ball of certain radius $R$ centred in this point. If the ...
2
votes
0answers
291 views

Gradient of an harmonic function

Let $ M $ be a Riemannian manifold and let $ f $ be an harmonic function on $ M $. By Unique continuation theorem we can assert that if $ \nabla f = 0 $ on an open subset $ \Omega \subset M $ then ...
14
votes
3answers
2k views

Geometrical interpretation of Ricci curvature

I see the scalar curvature $R$ as an indicator of how a manifold curves locally (the easiest example is for a $2$-dimensional manifold $M$, where the $R=0$ in a point means that it is flat there, ...
7
votes
3answers
404 views

Reference request for some topics in Differential Geometry like connections, metrics, curvature etc.

I wish to study some topics in Differential Geometry like Vector bundles, Metrics, Connections, Curvature,Geodesics, Holonomy, Characteristic classes etc. I am considering reading this book ' ...
12
votes
1answer
2k views

Are there simple examples of Riemannian manifolds with zero curvature and nonzero torsion

I am trying to grasp the Riemann curvature tensor, the torsion tensor and their relationship. In particular, I'm interested in necessary and sufficient conditions for local isometry with Euclidean ...
0
votes
1answer
72 views

Finsler metric, Find the fundamental form of F?

If $F= a+ k_1 b + k_2 b^2/ a$ , where $a$ is the Riemannian metric , $a = \sqrt{a_{ij} y^i y^j}$ and $b$ is the 1-form $b = b_i y^i$, then find the fundamental form $g_{ij}$ ?? I need the steps for ...
1
vote
1answer
84 views

Unit length tangent vectors on a Riemannian manifold

Let $X$ be a Riemannian manifold and $TX$ its tangent bundle. Is there a name for the $S^1$-bundle given by the unit length tangent vectors?