(Stub) The study of differential geometry with notions of infinitesimal distance given by a Riemannian metric. This allows us to consider questions about the shape of a manifold by studying its curvature.
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Uniformly quasi-isometric patches over classes of Riemannian manifolds
Suppose $(M^d,g)$ is a closed, connected Riemannian manifold.
Is there a constant $R > 0$ such that for all $z \in (M,g)$, for all $x \in B_R(z)$, \begin{equation} \frac{1}{2} \lVert \xi ...
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0answers
47 views
Are there more embeddings $U(2) \hookrightarrow SO(4)$?
It is easy to prove that $SO(4)$ acts transitively and freely on $S^2$ with fiber $U(2)$. Therefore, we can identify each point of $S^2$ with a particular embedding $U(2) \hookrightarrow SO(4)$.
My ...
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1answer
32 views
Space of embedded surfaces with a common point
Consider the space of all embedded orientable surfaces in $ R^3 $ of constant mean curvature (the minimal case is included) passing through the origin. I'm asking if there exists a topology on this ...
5
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0answers
46 views
What is the volume of Complex Projective Space with Fubini-Study Metric?
I try to compute the volume of the complex projective space $\mathbb{CP}^n$ with Fubini-Study metric, normalized to have diameter $=\pi/2$ i.e. the sectional curvatures lie between $1$ and $4$. Fix a ...
1
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1answer
37 views
Using Gauss-Bonnet to prove that geodesics have at most one point of intersection
Given an oriented Riemannian manifold $(M,g)$ of dimension $2$, such that $M$ has negative Gaussian curvature everywhere and $M$ is diffeomorphic to $\mathbb R^2$, I'm looking for a way to show that ...
2
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1answer
45 views
Prove Green formula
Let $(M^n,g)$ be an oriented Riemannian manifold with boundary $\partial M$.
The orientation on $Μ$ defines an orientation on $\partial M$. Locally, on the boundary, choose a positively oriented ...
2
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1answer
45 views
How to conclude $\frac{d}{dt}E(f)=-\int _M (\Delta f)\dot{f} d\mu$?
Can anyone explain to me how I can conclude $\frac{d}{dt}E(f)=-\int _M (\Delta f)\dot{f} d\mu$ by using integration by parts and $\langle f_1 ,f_2 \rangle_\mu:=\int_M f_1 f_2 d\mu$?
Where $M$ is a ...
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64 views
Exponential Map
this seems to be an easy question but I'm stuck anyway. Let $\Gamma$ be a submanifold of a Riemannian manifold $(M,g)$. Let further $U$ be a coodinate neighborhood
of $\Gamma$ such that a point ...
3
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0answers
20 views
structure of Riemannian manifold of isometries from C^n to C^m
Does anyone know a reference which gives the properties (geodesics, geodesic distance, etc) of the Riemannian manifold of isometries from $\mathbb{C}^n$ into $\mathbb{C}^m$, $m>n$, which map zero ...
4
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0answers
68 views
Solution to $\Delta_g u = \delta-1$ on a 2-sphere.
Let $S^2$ be the two-sphere, endowed with a Riemannian metric $g$, such that the volume of the sphere w.r.t. this metric is $4\pi$. Let $a \in S^2$. I am looking for an easy way to prove that the ...
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2answers
46 views
Is a Riemannian metric positive definite or positive semidefinite?
From Wikipedia
The Fisher information matrix is a N x N positive semidefinite symmetric matrix, defining a Riemannian metric on the N-dimensional parameter space,
But a Riemannian metric is ...
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0answers
40 views
Is there a relation between Super Riemannian manifolds and Kahler manifolds?
(This question has a physics motivation).
Could we establish any kind of relationship between Super Riemannian manifolds (super Riemannian structures on super manifolds) and Kahler manifolds, or at ...
5
votes
1answer
55 views
Zeta Regularized Determinant of Laplacian
Can anyone point me to a resource where the zeta regularized determinant of the Laplacian is explicitly computed for simple two dimensional surfaces, say a rectangle or torus or cylinder?
2
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0answers
50 views
Geodesics and Christoffel symbols
If these are satisfied then we are on a geodesic. Do I just need to plug in the condition given about the christoffel symbols and then see that the equations are allways fulfiled as long as $v=at+b$ ...
4
votes
1answer
73 views
Ricci tensor for a 3-sphere without Math packets
Let's have the metric for a 3-sphere:
$$
dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right).
$$
I tried to calculate Riemann or Ricci tensor's ...
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1answer
40 views
Orientability of $P_{\bf R}T{\bf RP}^{2n}$
I know the following fact :
(1) $ {\bf RP}^{2n}$ is non-orientable.
(2) $ {\bf RP}^{2n-1}$ is orientable.
(3) $P_{\bf R}T{\bf RP}^{2n}$ is orientable.
(4) $P_{\bf R}T{\bf RP}^{2n+1}$ ...
2
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1answer
73 views
Gradient in Riemannian manifold
I have a calculation involving a gradient and a parametrization, but I haven't been able to find out the relation between them. Let me explain.
Let $f:X↦R$ be a smooth function and $\mathrm{grad}f\in ...
2
votes
1answer
29 views
what is the inner product appeared in front of the integral?
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 ...
0
votes
1answer
40 views
Riemannian measure and Hausdorff measure in a general Riemannian Manifold
Let $ M $ be a Riemannian manifold and let $ \mu $ be its Riemannian measure. This is the measure obtained by Riesz reprersentation theorem such that for every continuous function with compact support ...
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1answer
32 views
Meaning of modulo diffeomorphism
I faced this sentence:
we consider the space of Riemannian metrics modulo diffeomorphism and scaling.
Can anyone explain to me what is the meaning of modulo diffeomorphism and scaling?
Thanks!
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0answers
33 views
Killing Vector Field determined by one point
I am trying to prove that if $X$ is a Killing vector field on a connected Riemannian manifold $(M,g)$ (i.e. $\mathfrak L_X g = 0$), then $X$ is determined by $X_p$ and $\nabla X|_p$ for any point $p ...
5
votes
2answers
88 views
Hopf's theorem on CMC surfaces
I got stuck reading the proof of the following theorem:
Theorem (Heinz Hopf) Let $X: S^2\to \mathbb R^3$ be a constant mean curvature immersion. Then $X(S^2)$ is a round sphere.
Proof: Let ...
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0answers
69 views
Lie bracket of vector fields definition equivalence
Lie bracket of vector fields is defined in two ways:
Let $\Phi^X_t$ be the flow associated with the vector field $X$, and let $d$ denote the
tangent map derivative operator. Then
the Lie ...
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0answers
74 views
Energy functional
During my study on Ricci Flow I faced some functional known as enery functional. For example Einstein-Hilbert functional is called an energy functional, also in Perelman's works ...
2
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1answer
36 views
Contraction of the second Bianchi identity
The second Bianchi identity is
$${R^a}_{b[cd;e]}=0$$
And contracting it with respect to $a$ and $e$ we get
$${R^a}_{b[cd;a]}=0 \Leftrightarrow $$
$${R^a}_{bcd;a}+R_{bc;d}-R_{bd;c}=0$$
What I don't ...
2
votes
1answer
78 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 ...
5
votes
2answers
126 views
Justification for this manipulation in a proof of the first variation of energy formula
As a part of my current homework assignment, I am to derive the first variation of energy identity. Working out the problem with my friends, we came to exactly the same argument as presented in these ...
2
votes
1answer
25 views
Cut locus of $\mathbb{CP}^n$
I can show that the cut locust of some $p\in\mathbb{RP}^n$ is just a copy of $\mathbb{RP}^{n-1}$ coming from an equatorial $S^{n-1}$ sphere under the projection $S^n\mapsto\mathbb{RP}^n$.
I know that ...
0
votes
0answers
67 views
Volume element of the Sphere
If we consider the sphere on $E^3$ with Riemannian metric $G=dx + dy + dz$ then transforming to spherical coordinates we get $G=R^2 d{\theta} +R^2 sin(\theta) d{\phi}^2 $.
Hence the volume form is ...
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1answer
40 views
Volume of a 3D sphere of radius $R$ using Riemannian metric in stereographic coordinates
The question is pretty much in the title. We were also given the hint that it could be useful to use spherical coordinates when calculating the integral (the actual answer is not required, just its ...
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0answers
30 views
Non-linear transformation preserving stereographic Riemannian metric on the sphere of radius R
I have been given a the Riemannian metric of a sphere of radius R in stereographic coordinates:
$$G=4R^4\frac{du^2+dv^2}{(R^2+u^2+v^2)^2}.$$ I have shown that this metric is preserved under rotation, ...
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1answer
72 views
Quotient theorem for tensors
Can somebody please explain to me how the following statement is true?
The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
4
votes
0answers
40 views
Metric on Steifel and Grassmannian manifolds generalizing Fubini-Study
If $F$ is $\mathbb{R}, \mathbb{C}$, or $ \mathbb{H}$, the Grassmannian manifold $G_k(\textbf F^n)$ is the space of all $k$ dimensional subspaces of the $n$ dimensional vector space $F^n$. The Stiefel ...
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1answer
40 views
Confusion regarding uniqueness of Levi-Civita connection
Assuming a Levi-Civita connection exists it is uniquely determined.
Using $\nabla g = 0$ and the symmetry of the metric tensor $g$ we
find:
$ X (g(Y,Z)) + Y (g(Z,X)) - Z (g(Y,X)) = ...
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 ...
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0answers
63 views
Riemannian curvature and its application on covariant derivative of tensors
This identity can be generalized to get the commutators for two
covariant derivatives of arbitrary tensors as follows $ \begin{align} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s ; ...
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0answers
44 views
Existence of Solution: Embedding from 2D Euclidean space to a circle
Given a real matrix $X$ with $n$ rows and 2 columns, can the matrix be transformed to a real matrix $Y$ such that all the points formed by the rows of $Y$ lie on a circle (2d) and their inter-point ...
0
votes
1answer
31 views
Question regarding Nash-Kuiper embedding theorem
In Wikipedia description of Nash-Kuiper theorem, it says:
Let $(M,g)$ be a Riemannian manifold and $f: M^{m} \rightarrow \mathbb{R}^n$ a short $C^{\infty}$-embedding into Euclidean space
...
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0answers
43 views
Reason for defining Riemannian curvature tensor and torsion tensor in particular way
I saw how Riemannian curvature tensor and torsion tensor are defined, but I am not sure why they are defined that way. In 3-dimensional euclidean space with ordinary multivariable calculus, the ...
4
votes
1answer
86 views
Problem about Ricci Flow
On page 12 of "Lectures On Ricci Flow" by Peter Topping is written:
In two dimensions, we know that the Ricci curvature can be written in
terms of the Gauss curvature $K$ as $Ric(g) = Kg$. Working ...
4
votes
1answer
53 views
covariant derivative vs. exterior derivative
I have the following question. Let $M$ be a Riemannian manifold with metric $g$ and $\nabla$ the Levi-Civita connection. Let furthermore $\alpha \in \Omega^{k}(M)$ be a $k$-form such that $\nabla ...
2
votes
1answer
58 views
the Ricci curvature in two dimension
In two dimensions, we know that the Ricci curvature can be written in
terms of the Gauss curvature K as $Ric(g) = Kg$.
Can anyone prove this?
Sorry if the question is too trivial :).
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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 ...
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2answers
50 views
Showing that the riemanian metric $\frac{g}{\sqrt{x^2+y^2+z^2}}$ is complete
I would like to show that a certain Riemannian metric defined on $\mathbb{R^3}$ is complete. The metric is given in the following sentence from this article (pg 160):
... a Riemannian metric on ...
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1answer
36 views
Elementary definition: what's a parallel volume-form?
This is a very elementary question,
What is the definition for a volume form (or $n$-form) to be parallel with respect to the metric?
To find out more about the concept, what kind of topic do I need ...
1
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1answer
95 views
Curvature of a metric defined on an open disc in $\mathbb{R}^2$
Let $D$ be an open disc centred at the origin in $\mathbb{R}^2$. Give $D$ a Riemannian metric of the the form $(dx^2+dy^2)/f(r)^2$, where $r=\sqrt{x^2+y^2}$ and $f(r)>0$. Show that the curvature of ...
1
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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 ...
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0answers
23 views
Linear Operators on a Riemannian Manifold
Given an $m$-dimensional connected Riemannian manifold $(M,g)$ with Levi-Civita connection $\nabla$, we consider a symmetric $(0,2)$-tensor $A$ with $\nabla A=0$. There is an associated $(1,1)$-tensor ...
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1answer
63 views
Parallel transport for a conformally equivalent metric
Suppose $M$ is a smooth manifold equipped with a Riemannian metric $g$. Given a curve $c$, let $P_c$ denote parallel transport along $c$. Now suppose you consider a new metric $g'=fg$ where $f$ is a ...
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0answers
35 views
A $k$-form is thought of as measuring the flux through an infinitesimal k-parallelepiped
On the wikipedia has written "A $k$-form is thought of as measuring the flux through an infinitesimal k-parallelepiped." How does a $k$-form do this? if this sentence is right, then the flux of which ...
