Tagged Questions

2
votes
1answer
79 views

Gradient of a functional

Given a compact manifold with a Riemannian metric $g$, we define the total scalar curvature by $$E(g)=\int_M RdV$$ Let us consider the first variation of $E$ under an arbitrary change of metric. We ...
4
votes
2answers
109 views

geometric interpretation of Lie bracket

On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written: We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to which the integral ...
0
votes
1answer
72 views

About Sectional Curvature

In a paper by Yann Ollivier: Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel transport of $w_x$ from ...
1
vote
1answer
81 views

Difference between “Live” and “Define”

In many mathematical text to determine an object on manifold, the verbs "live" and "define" are used. I'm interested to know whether there is a difference between the concepts of "to define" and "to ...
4
votes
1answer
108 views

Why do we need Lie derivative?

If a manifold is equipped with Levi-Civita connection, Why do we need Lie derivative? In Euclidean space to calculate directional derivative of a vector field V along W, we parallel transport V along ...
5
votes
1answer
72 views

A question about concept of pushforward

In An Introduction to Smooth manifolds by Lee is written: for any smooth vector fields V and W on a manifold $M$, let $\theta$ be the flow of $V$, and define a vector $(\mathcal{L}_v W)_p$ at each ...
2
votes
2answers
113 views

Normal coordinates

Let $M$ a riemannian manifold and $\nabla$ the Levi-Civita conection. Ineed to prove the next. Let $B$ an open ball of radius $r$ in $T_pM$ such that $exp_p\mid _B$ be a difeomorphism over an open $U$ ...
1
vote
1answer
68 views

Show that the vector field $\operatorname{grad}f$ is smooth

Let $M$ be a Riemannian manifold and $ f:M\rightarrow\mathbb{R}$ be a smooth function. Define a vector field $\operatorname{grad}f$ in $M$ as $$\langle\operatorname{grad}f,\,V\rangle=df(V)$$ for all ...
5
votes
1answer
233 views

Vector field and integral curve

Let M a riemaniann manifold, $V$ a vector field in M, and $\phi_t$ the respective flow Let $p\in M$, and $\gamma$ the integral curve of $V$. Show that for all $v\in T_pM$ ...
0
votes
0answers
42 views

Riemannian measure

Let $ M $ be a compact Riemannian manifold and let $ g,h $ two Riemannian metric on $ M $. Moreover, assume that $ \mu_g $ and $ \mu_h $ be the Riemannian measures induced by $ g $ and $ h $. Now let ...
3
votes
0answers
35 views

A Simons' type inequality

I have a problem with the inequality (5) in the article 'Estimates for stable minimal surfaces in three dimensional manifolds' of R.Schoen. As the author suggests this inequality comes from 'well ...
5
votes
1answer
87 views

Diffeomorphic riemannian manifolds and volume forms

Maybe the question will be stupid, but I'm a beginner in riemannian geometry... We have two riemannian manifolds $(M,g)$, $(\overline M,\overline g)$ and a diffeomorphism $F:M\rightarrow\overline M$ ...
2
votes
1answer
65 views

Open sets of the tangent bundle in a Riemannian manifold

Let $M$ be a Riemannian manifold with a metric $g$ and $(U,\varphi)$ a chart around a point $p\in M$. By a Remark page 63 of Riemannian Geometry by M. Do Carmo, it seems that any open set ...
25
votes
2answers
590 views

PDEs on manifold: what changes from Euclidean case?

I know some PDE theory for nice open domains in $\mathbb{R}^n.$ I want to know what the changes are when I switch to other domains like manifolds. For example, do things like Poincare's inequality ...
5
votes
1answer
225 views

Uniformization Theorem for compact surface

Why in proof of proposition 6 of http://arxiv.org/abs/0909.1665, they claim that if a embedded surfaces $\Sigma^2 \subset (M^3,g)$ is homeomorphic to $\mathbb{RP}^2$, where $M$ is compact manifold, ...
5
votes
1answer
71 views

Isometries from Diffeomorphisms

Given a compact, connected Lie group G of diffeomorphisms on a manifold M, how to construct a Riemannian metric on M such that elements of G are isometries of M?
1
vote
0answers
30 views

Minimum is attained in a subset of a Sobolev space

Let $\Omega \subset \mathbb R^n$. I have a functional of the form, $$\int_{\Omega}f(x,u,\nabla u)dx$$ where $u \in W^{1,p}(\Omega, M)$, $M \subset \mathbb R^d$ is a compact, smooth Riemannian ...
2
votes
1answer
138 views

Levi-Civita connection

Well if $\Sigma$ is a submanifold of $R^{n+p}$ and $\{e_i,e_\alpha\}$ is orthonormal frame over $\Sigma$ where the $e_i$'s are tangent and the $e_\alpha$'s are normal to $\Sigma$. Can anyone prove ...
9
votes
2answers
420 views

Is the Nash Embedding Theorem a special case of the Whitney Embedding Theorem?

The Whitney Embedding Theorem states that every smooth manifold can be embedded in Euclidean space. The Nash Embedding Theorem states that every Riemannian manifold can be embedded in Euclidean ...
2
votes
0answers
37 views

Complex structure on the product of two complex Kaehler manifolds

I have the following question: Let $(M,J_{M}, \omega_{M})$ and $(N,J_{N}, \omega_{N})$ be two Kaehler manifolds. I consider $M \times N$. Can I introduce on $M \times N$ a complex structure ? I think ...
0
votes
1answer
103 views

cotangent bundle splits as a product?

Let $M$ and $N$ be two Riemannian manifolds with Riemannian metrics $g$, $h$ respectively. We consider the product $M \times N$ with metric $g \oplus h$. By the metric we get an isomorphism of bundles ...
4
votes
2answers
145 views

orthonormal vector fields

In a Riemannian manifold $(M^n,g)$, is it true that given a point $p$ we can find $n$ vector fields $X_i$ on a neighborhood $U$ of $p$ such that $g(X_i,X_j)(x)=\delta_{ij}$ with $x \in U$? It seems ...
3
votes
2answers
186 views

Does a diffeomorphism between manifolds induce an isomorphism of Sobolev spaces?

Let $M$ be a Riemannian manifold, and define the Sobolev spaces $H^k(M)$ to be the set of distributions $f$ on $M$ such that $Pf \in L^2(M)$ for all differential operators $P$ on $M$ of order less ...
0
votes
0answers
44 views

Relation with Jacobi fields in a small neighbourhood of some point in a complete manifold

I have the following question: Let $(M,g)$ be a complete Riemannian manifold with all sectional curvatures non-positive. Let $p \in M$ and consider the function $d(x)=dist_{g}(x,p)$ in a ...
6
votes
3answers
298 views

Why are we interested in closed geodesics?

There's a lot of work about the existence, number and other properties of closed geodesics on a Riemannian manifold (belonging to some specific class of manifolds). In the case of geodesics ...
0
votes
1answer
62 views

Defintion of totally geodesic flat submanifold

I don't know if this is an inappropriate question to post on stackexchange, but could somebody give me (reference me) a precise definition of "totally geodesic and flat submanifold" of a riemannian ...
2
votes
0answers
109 views

Extending Tensor Fields defined on Manifolds to Ambient Space

I am currently reading about tensor fields on manifolds, and I came across two comments that sound contradictory to me. The first comment is made in the book by James Munkres "Analysis on Manifolds", ...
2
votes
2answers
108 views

help on connections

In the book on Riemannian Geometry by John Lee ("Riemannian manifolds: an Introduction to Curvature") the author gives an exercise on page 54 involving connections: Let $\triangledown$ be a linear ...
1
vote
1answer
102 views

Riemannian metric - basic question

I tried to google the following but couldn't find an answer that helped - so I hope I might find some here - the question is short and very basic (I guess) : what does it mean when someone writes ...
6
votes
1answer
371 views

Isometries preserve geodesics

Let $f$ be an isometry (i.e a diffeomorphism which preserves the Riemannian metrics) between Riemannian manifolds $(M,g)$ and $(N,h).$ One can argue that $f$ also preserves the induced metrics $d_1, ...
6
votes
2answers
267 views

Each point of $M$ has a smooth coordinate neighborhood in which the coordinate frame is orthonormal iff $g$ is flat

Let $(M,g)$ be a Riemannian manifold. Then I want to show that these are equivalent: (i) Each point of $M$ has a smooth coordinate neighborhood in which the coordinate frame is orthonormal. ...
1
vote
1answer
72 views

Is $\Gamma_{jk}^{t}=\frac{1}{2}g^{it}g_{ij,k}$?

Here $\Gamma_{jk}^{t}$ is the Christoffel symbol of the second kind, and $g$ is the Riemann metric on a Riemann manifold.When learning Riemann Geometry, we are usually introduced to the following ...
3
votes
1answer
58 views

Products of LCF manifolds

Is the Cartesian product of two Weyl-flat manifolds Weyl-flat as well? Here, by "Weyl-flat" I mean that the Weyl tensor of the metric vanishes everywhere. I know that the product of space forms isn't ...
1
vote
1answer
429 views

Proving some basic properties of covariant differentiation

I have the following somewhat awkward definition of covariant differentiation along a curve: Let $S \subseteq \mathbb{R}^N$ be a smoothly and isometrically embedded manifold, and $\alpha : I \to ...
0
votes
0answers
107 views

Transportation of a tangent vector by left-invariant translation in complete Riemannian manifolds

How can i guarantee that a transportation of a vector, defined on the tangent space at a element in complete Riemannian manifold, by left-invariant translation is a parallel transport along geodesic ...