1
vote
0answers
16 views

Fundamental solution of Laplacian on manifold

I'm looking for a reference for the result that there exists a fundamental solution for the Laplacian on a flat torus $$\Delta \Gamma(x-y) = \delta(x-y), \quad x,y \in \mathbb T^2.$$ and that, ...
0
votes
0answers
14 views

Bianchi identity number of independent equations

What is the number of independent equations of the second Bianchi identity: $$R_{abcd;e}+R_{abec;d}+R_{abde;c}=0$$
3
votes
0answers
26 views

If $\operatorname{Def} X = \frac{1}{2} \mathcal L_X g$ then what is $\operatorname{Def}^* \operatorname{Def} X$?

Let us define a deformation operator $\operatorname{Def}$ on a Riemannian manifold $(M,g)$, acting on divergence-free vector fields as $$ \operatorname{Def} X = \frac{1}{2} \mathcal L_X g, $$ where ...
4
votes
1answer
43 views

Riemann curvature product metric

Suppose that $M=M_1 \times M_2,$ with the product metric $g= g_1 \oplus g_2.$ Let $p\in M$ and suppose that $X \in T_pM_1$ and $Y\in T_pM_2.$ I want to show that $R(X,Y,Y,X)=0,$ at the point $p.$ I ...
2
votes
2answers
46 views

Topologies in a Riemannian Manifold

I'm studying Differential Manifolds using Manfredo do Carmo's Book (Riemannian Geometry) and although I see no mention of this in Do Carmo's book, it's really easy to see a Riemannian Manifold as a ...
1
vote
1answer
36 views

A simple metric question

In their article Killing vector fields of standard static spacetimes, Dobarro and Unal derived the following simple identity. Note that if $h:I→R$ is smooth and $Y,Z∈ {\frak{X}}\left(I\right)$, ...
1
vote
0answers
34 views

The uniqueness of the Einstein metric on $\mathbb CP^n$

Is the Fubini-Study metric the unique Einstein metric (up to scaling by a constant) on $\mathbb CP^n$? More restrictively, Is the Fubini-Study metric the unique Kahler-Einstein metric (up to scaling ...
0
votes
1answer
34 views

Prove that there are no complete regular minimal surfaces lying above a paraboloid

Prove that there are no complete regular minimal surfaces lying above a paraboloid contained in $U=\{(x,y,z) \in \mathbb{R}^3 : a(x^2+y^2)<z\}$. Here $a>0$. I've had this problem on my mind ...
3
votes
1answer
42 views

Question about complete metric on manifolds

I've recently been wondering about whether non-complete metrics on manifolds can be transformed into complete metrics on manifolds and whether all manifolds have complete metrics. After some googling ...
1
vote
0answers
26 views

Ricci scalar algebra

Is this derivation correct? If $R=0$, $R$ being the scalar curvature, then: $$R_{;k}=0$$ $$(g^{ac}g^{bd}R_{abcd})_{;k}=0$$ $$g^{ac}g^{bd}(R_{abcd})_{;k}=0$$ $$R_{abcd;k}=0$$
1
vote
1answer
62 views

Exponential map and distance on a Riemannian Manifold

I'm trying to solve an exercise which I thought at first seemed simple but I'm having some trouble pegging down the error term. The question is to prove that on a Riemannian manifold we have ...
2
votes
1answer
34 views

Conformal classes and almost-complex structures

It is well-known that on closed oriented surfaces $S$, conformal classes of metrics on $S$ correspond bijectively to complex structures on $S$. My understanding is that this correspondance goes as ...
3
votes
3answers
104 views

Differentiability of the distance function

Suppose that $d:X \times X \to \mathbb{R}$ is a geodesic distance function on a smooth Riemannian manifold $X$ ($d$ is determined by metric tensor) and $x \in X$ is fixed. What can be said about the ...
3
votes
1answer
36 views

The isometry group of the simply-connected Ricci-flat closed manifold

If $M$ is a simply-connected Ricci-flat closed manifold, then is $I(M)$ the isometry group finite?
1
vote
1answer
59 views

Distance between two points on the Clifford torus

How can I obtain the distance between two points $\mathbf{x}=(x_1,x_2,x_3,x_4)$ and $\mathbf{y}=(y_1,y_2,y_3,y_4)$ that belong to the $2$-torus $\mathbb{S}^1\times \mathbb{S}^1$? This is, I want to ...
3
votes
0answers
45 views

Ricci SCALAR curvature

Are there any manifolds that have NULL SCALAR curvature but not null ricci curvature tensor? For dim>2, obviously. Edit: Are there, also, any manifolds of null scalar curvature but ricci curvature ...
1
vote
1answer
40 views

A question on self-dual differential 2-forms

This question is from Lemma 2 in Derdzinski's paper. Let $$\omega=e_1\wedge e_2+e_3\wedge e_4, \eta=e_1\wedge e_3+e_4\wedge e_2, \theta=e_1\wedge e_4+e_2\wedge e_3$$ be a basis for self-dual ...
0
votes
0answers
39 views

The curvature of surfaces in Euclidean space (Theorema Egregium)

The below animation is from Wikipedia. It shows how a helicoid can be deformed into a catanoid and vice versa without stretching. Because of this, the Theorema Egregium shows that the Gaussian ...
2
votes
0answers
78 views

Pushforward along an exponential map

My differential geometry is a bit rusty, so I'd like some help with what follows: I have the following setting: $M,N$ Riemannian manifold of dimension $m<n$ with codimension $d$, $M$ in embedded ...
3
votes
2answers
65 views

Is there an easy way to reason about expressions involving lots of indices?

I have been reading some Riemannian geometry recently. So far, I think I am understanding the concepts well enough. However, I am finding it difficult to translate some of the notation into meaning. ...
7
votes
1answer
103 views

Is there a codifferential for a covariant exterior derivative?

For forms on a Riemannian $n$-manifold $(M,g)$ there is a notion of a codifferential $\delta$, which is adjoint to the exterior derivative: $$\int \langle d \alpha, \beta \rangle \operatorname{vol} = ...
5
votes
1answer
58 views

Naturality of the pullback connection

I'm completely stuck proving the naturality of the pullback connection. The strategy suggested is a follows: We let $\phi: (M,g) \to (\tilde{M}, \tilde{g})$ be an isometry, with connections ...
1
vote
1answer
46 views

Noncomplete riemannian manifolds

In Lee's Riemannian manifolds text, he claims that "on $\mathbb{R}^n$ with metric $(\sigma^{-1})^* g$ obtained from the sphere from stereographic projection, there are geodesics that escape to ...
1
vote
1answer
51 views

Boothby's exercise $I.3.1$ on showing that a subset of $\mathbb{R}^{2}$ is not a manifold

First time here, so sorry for the rookie mistakes. I'm a $4^{th}$ year physics student taking Riemannian Geometry so my background on the subject is very small. I'm trying to solve Boothby's exercise ...
3
votes
1answer
60 views

If $\operatorname{div} X = 0$ what can be said about $X^\flat$?

If vector field $X$ is divergent free $$\operatorname{div} X = 0$$ what are the properties of a corresponding covector field $X^\flat$ (via musical isomorphism with a metric $g$)? Are there some ...
2
votes
2answers
58 views

Christoffel's symbols for a dual connection

Suppose that $\Gamma^{\beta}_{i\alpha}$ are Christoffel symbols for a connection with respct to a (local) basis $\{E_1,...,E_n\}$. I tried to prove that the Christoffel symbols for a dual connection ...
1
vote
0answers
38 views

Computing the Fubini-Study metric

I would like to compute the Fubini-Study metric $g_{FS}$ for $\mathbb{CP}^{n}$ using as definition the metric induced from the round metric of the sphere by the Hopf fibration. I tried to compute on ...
0
votes
1answer
37 views

When to take derivative with respect to distance?

I had a previous question about the divergence in spherical coordinates and using the usual formula found on wikipedia "List of formulas in Riemannian geometry" I could not get the correct form of the ...
1
vote
0answers
49 views

Norm equivalence of p-forms

Let $U$ be a bounded domain in the Euclid space $(\mathbb{R}^d,g)$. $g_{\wedge^p}$ denotes the fiber metric of $\wedge^pT^{\ast}\mathbb{R}^d$ derived from $g$. $A^p$ denotes the set of p- forms on ...
0
votes
0answers
29 views

Relationship between differentiation and integration of vector fields?

Let $V\in\Gamma(T\mathbb{R}^n)$ be a vector field and $\gamma:[a,b]\to \mathbb{R}^n$ a curve. Let $\nabla$ be the Euclidean connection, i.e. $\nabla_XY=XY^k\frac{\partial}{\partial x^k}$. We have a ...
1
vote
1answer
56 views

Why are parallel vector fields called parallel?

In Lee's "Riemannian Manifolds: An Introduction to Curvature" given a curve $\gamma:[a,b]\to M$ and a tangent vector $V_0\in T_{\gamma(t_0)}M$, where $t_0\in [a,b]$, there is a drawing of the parallel ...
2
votes
0answers
101 views

Confusion regarding Riemann normal coordinates

I'm trying to understand Riemann normal coordinates. This "simple" example using the surface of a unit sphere is from http://www.maths.bris.ac.uk/~macpd/gen_rel/snotes.pdf (p26). The “north pole” ...
0
votes
1answer
35 views

Riemannian geometry algebra

Is this derivation correct? $$ R^{ab}_{;a}=0 $$ $$ g_{ac}g_{bd}R^{ab}_{;a}=0 $$ $$ (g_{ac}g_{bd}R^{ab})_{;a}=0 $$ $$ R_{cd;a}=0 $$ And does that mean I now have $n^3$ equation as opposed to $n$?
2
votes
0answers
31 views

mean curvature of a nonlinear parabolic PDE

I want to compute the mean curvature of a reaction diffusion equations of the form $$ u_t=u_{xx}-u^2 $$ Any suggestions is appreciated!
4
votes
1answer
59 views

Space of oriented lines in $\mathbb{R}^{n+1}$ as symplectic quotient.

I've been working out a nice example of symplectic reduction, and have come to a solution only after quite a lot of effort. So I was wondering if anyone knew a more straightforward route to the ...
0
votes
1answer
33 views

Connected and simply connected neighborhoods

Suppose that $E \to M$ is a (smooth) vector bundle over smooth manifold $M$. One can find the covering $\{U_i\}_i$ with the property that $E|_{U_i}$ is trivial vector bundle. The prooblem is the ...
3
votes
1answer
71 views

Why Riemannian metrics have to be smooth?

Why do Riemannian metrics have to be smooth? Can you give an example of a smooth curve with a none smooth metric and show me what possibly will go wrong if our metric is not smooth?
1
vote
1answer
27 views

A Couple Formulas in Besse's “Einstein Manifolds”

In Besse's "Einstein Manifolds," Chapter 6D, there are 2 formulas which I am interested in, which apply to compact Riemannian $4$-manifolds: ...
0
votes
1answer
37 views

Simple question on symmetric tensors 2

This question is related to this one Simple question on symmetric tensors. To prove that a vector field $Z$ is Killing, we use the identity $$0=(L_Zg)(X,Y)=g(X,\nabla_YZ)+g(\nabla_XZ,Y)\ \ \ \forall ...
1
vote
1answer
20 views

Is the image under a homeomorphism of the cut locus $C_p$ a null-set?

Let $M$ be a complete Riemannian manifold with a point $p  \in M$ and let $U \subset T_pM$ be an open disk containing $0_p$ in the tangent space to $p$. By $C_p$ we denote the image of the boundary of ...
2
votes
1answer
51 views

Hessian is proportional to the metric everywhere

Let $(\Omega^{n+1},g)$ be a compact Riemannian manifold with smooth boundary. Let $f\in C^{\infty}(\bar{\Omega})$ satisfies $\operatorname{Hess}f=\frac{1}{n+1}g.$ Suppose the minimum of $f$ occures at ...
1
vote
1answer
100 views

The relation between geodesics and distances on a Riemannian manifold

My question is about computing the distance between two points in a Riemannian manifold. Suppose that $(M,g)$ is compact so that it is geodesically complete and geodesically convex. Let ...
2
votes
1answer
31 views

In uniform circular motion in R^2, is acceleration in the normal bundle?

In physics we learn that accleration is a vector quantity parallel to the radius and orthogonal to the velocity. With the embedding $\mathbb{S}^1 \hookrightarrow \mathbb{R}^2$ and the induced ...
0
votes
1answer
38 views

Definition of a lipschitz 1-form on a manifold

What is the definition of a Lipschitz-regular 1-form on a riemannian manifold?
1
vote
0answers
51 views

geometrical consequences of nonpositive or negative Ricci curvature.

well,that's pretty much the question. I'd like to know if somebody could point me out if there's any geometrical implications following an upper bound on the Ricci curvature on a riemannian manifold. ...
2
votes
0answers
26 views

The invariance of the Ricci tensor under diffeomorphisms and its non-ellipticity.

Consider $(M,g)$ a compact Riemannian manifold. When viewed as a second order (non-linear) differential operator $$ \text{Ric} : C^{\infty}(\text{Sym}^2_+T^*M) \to C^{\infty}(\text{Sym}^2T^*M), $$ the ...
2
votes
0answers
53 views

Ovals of constant $ k_g$ on constant $K$ surfaces

Prove that: Constant geodesic curvature lines on constant Gauss curvature surfaces are closed Ovals/Loops. Find perimeter/length of this Oval/Loop in terms of $ k_g$ and $K$ I believe the proof ...
1
vote
0answers
47 views

Manifold characteristics in terms of Riemannian metric

I wonder what characteristics of Riemannian manifold can be expressed in terms of metric? Are there any results similar to Gauss–Bonnet theorem? Does the Riemannian metric give any information about ...
1
vote
1answer
48 views

surface in $R^3$ that has $ds^2 = du^2/v^2 + dv^2/v^2$

For a 2D surface, if we have the first fundamental form of $$ ds^2 = du^2/v^2 + dv^2/v^2$$, can we integrate it out to get the parameter form of the surface embedded in $R^3$? I tried something like ...
0
votes
0answers
67 views

Constant curvature geodesic circles on a surface with constant Gauss curvature

Referring to: Curvature of geodesic circles on surface with constant curvature, Is it possible to combine further the last two of the three equations in the link given above into a single ODE / PDE ...