(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.
3
votes
1answer
76 views
A question about laplacian of the second fundamental form
Let $ f:M \rightarrow N $ be an immersed oriented hypersurface, $ e_{1}, \ldots e_{n},e_{n+1} $ be an orthonormal frame of $ N $ such that $ e_{1} \ldots e_{n} $ is an orthonormal frame of $ M $. Let ...
3
votes
1answer
310 views
Commutation formula for covariant derivative
Suppose $\nabla$ is the Levi-Civita connection on Riemannian manifold $M$. $X$ be a vector fields on $M$ defined by $X=\nabla r$ where $r$ is the distance function to a fixed point in $M$. $\{e_1, ...
1
vote
1answer
81 views
Isometry and geodesic
Let $F: M \rightarrow N$ an isometry and $M,N$ two riemannian manifold. How can I prove that the set of fixed points of F isometry (among riemannian manifold) is a geodesic? In general is it a curve?
0
votes
0answers
75 views
Tensor calculus solution-why?
The text I read says that $\displaystyle\frac{\partial^2
x^\alpha}{\partial x^\delta
\partial x^\gamma}\frac{\partial x^\delta}{\partial
x^\beta} = 0$ leads to the solution $x^\alpha =
...
3
votes
1answer
139 views
Geodesics on a 2-sphere
I've been doing some work where I need to find the geodesics in a given Riemannian Manifold. Let's take the example of the two sphere, for simplicity, with unitary radius. The distance between two ...
1
vote
0answers
149 views
How to derive covariant derivative and Lie derivative of tensors
1) As title says, how does one derive the following
equation for covariant derivate of tensor:
$A^{\alpha}{}_{;\beta} = A^{\alpha}{}_{,\beta} +
\Gamma^{\alpha} {}_{\gamma\beta}A^\gamma$
where ...
2
votes
1answer
77 views
Is a covering space of a manifold second countable?
I have been told that the covering space of a manifold is again a manifold. Let $f :X\rightarrow Y$ be a covering map and $Y$ is a $n$-manifold. It is easy to show that $X$ is a locally euclidean and ...
3
votes
0answers
103 views
Intuition for Fisher information metric
In statistical maniolds $S=\{p_\theta\}$,$\theta=(\theta_1,\dots,\theta_n)$, the Riemaanian metric usually defined is the Fisher information metric $$g_{ij}(\partial_i,\partial_j)=\int \partial_i(\log ...
1
vote
0answers
24 views
A question about an estimate in $ C_\infty $ of the Gauss map
Let $ g:\Omega \rightarrow C \cup \infty = C_\infty $ be the gauss map of a conformal minimal immersion $ X: \Omega \rightarrow R^3 $. We know that: $ ...
6
votes
2answers
166 views
Elliptic estimates on compact manifolds
Hey where may I find elliptic estimates for PDEs on compact (no boundary) Riemannian manifolds? I want a source/paper/book where I can cite it.
For example, for $L$ a linear elliptic operator, (eg. ...
2
votes
0answers
43 views
Lie derivative of curvature
Let $M$ be a Kahler manifold, with Kahler metric $g$. Let $X$ be a holomorphic Killing vector field of $g$, i.e. $\mathcal{L}_{X} g = 0$, where $\mathcal{L}_{X}$ is the Lie derivative along $X$. Let ...
2
votes
1answer
50 views
Existence of vector fields
Does there exists two vector fields $X$ and $Y$ on $\mathbb R^2$ such that the following are satisfied?
$X(0)= Y(0)= 0$, where $0\in \mathbb R^2$ and for others points
$q\in \mathbb R^2$, we ...
3
votes
0answers
88 views
Is $\bar{\partial}_E + \bar{\partial}_E^*$ a Dirac operator?
I have previously asked about Weitzenböck identities and received some great answers on MathOverflow.
One question which has arisen from the post is the following:
Let $E$ be a hermitian ...
2
votes
1answer
37 views
Schoen Estimates (part 3)
I'm referring to the article 'Estimates for stable minimal surfaces in three dimensional manifolds' of Richard Schoen
In the first paragraph of the proof of theorem 2 the author seems to assert that ...
3
votes
2answers
88 views
Divergence theorem on Hyperbolic space
Given a vector field, say $F$, defined on a manifold $U$, the divergence theorem states that:
$$\int_U\nabla \cdot F dV=\int_{\partial U} F d \Sigma .$$
Well if the manifold is $\mathbb R ^n$ and $F$ ...
3
votes
0answers
51 views
Schoen curvature estimates (part 2)
I'm referring to the article ''Estimates for stable minimal surfaces in three dimensional manifolds'' by Richard Schoen.
I have a question about the proof of theorem 1. In the last step of the proof ...
1
vote
0answers
103 views
Riemannian metric for euclidean geometry
I am a beginner to Riemannian geometry. Following is my question.
In the Euclidean space, say $\mathbb{R}^3$, let us consider a plane, for simplicity, say one passing through the origin, ...
1
vote
1answer
54 views
Regular compact domains of a Riemannian manifolds
In a Riemannian manifold $ M $ a regular compact domain $ D $ is a compact subset of $ M $ with non empty interior and such that for every $ p \in \partial D $ there exists $ \left(U,\varphi\right) $ ...
1
vote
1answer
44 views
A question about an estimate
Let $ f:M \rightarrow N $ be a minimal immersion where $ M $ is a compact two dimensional manifold and $ N $ is a three dimensional manifold. Let $ |A|^2 $ be the square of its second fundamental ...
1
vote
1answer
61 views
Thoughts about sectional curvature
I'm currently trying to understand the sectional curvature of riemannian manifolds and I don't know if I'm thinking correctly.
So, say we have a riemannian manifold $(M,g)$ with constant sectional ...
1
vote
0answers
59 views
$|\Delta f|^2$ in local coordinates
The Laplace-Beltrami in local coordinates (for hypersurfaces in my case) is
$$\Delta f = \frac{1}{\sqrt {|g|}} \partial_i \left(\sqrt{|g|} g^{ij} \partial_j f \right)$$
Is there a nice formula for its ...
0
votes
2answers
76 views
Constant Riemannian Metric
Let $M=\mathbb R^n$ and define for each $x\in \mathbb R^n$, define
$$\langle v,w\rangle_x= \langle v,w\rangle_0$$
where $v,w\in T_x\mathbb R^n\equiv \mathbb R^n\equiv T_0\mathbb R^n$. Hence we see ...
3
votes
1answer
106 views
Is there a non-variational derivation of Snell's law from Fermat's principle?
Every proof I've seen of Snell's law from Fermat's principle uses some sort of variational argument, mostly involving variational calculus. Niven's wonderful book, Maxima and Minima Without Calculus ...
4
votes
1answer
73 views
Manifold without conjugate points and positive curvature
I'm looking for an example of a complete riemannian manifold with sectional positive curvature and without conjugate points. I've tried the projective space, but the identfication used to construct it ...
3
votes
1answer
51 views
Riemannian metric. Help with notation.
I was just reading about the hyperbolic space (upper-half plane model) and i'm getting kind of
confused about the notation for the Riemannian metric. The half-plane is defined as
$$
H = \{(x,y) \in ...
3
votes
2answers
150 views
Riemannian Geometry book to complement General Relativity course?
What would be a good Riemannian Geometry (or Differential Geometry) book that would go well with a General Relativity class (offered by a physics department)? I'm in one right now, but I'd like a pure ...
2
votes
0answers
59 views
The standard connection in a regular surface is symmetric
The concrete setting: Let $M\subset \mathbb{R}^3$ be a regular surface. A vector field $X$ in $M$ is a differentiable function $X: M\to \mathbb{R}^3$ such that $X(p)\in T_pM$.
Here we are taking the ...
3
votes
1answer
60 views
What are all isometry classes of the 2-sphere?
In topology, one learns how to classify the compact surfaces up to homeomorphism. And in fact, since "homeomorphic" and "diffeomorphic" coincide in dimension 2, we can classify the compact (smooth) ...
6
votes
3answers
144 views
Are the Ricci and Scalar curvatures the only “interesting” tensors coming from the Riemannian curvature tensor?
In studying Riemannian geometry, one learns about the Riemann curvature tensor
$$Rm(X,Y,Z,W) = \langle \nabla_X\nabla_YZ -\nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z, W\rangle$$
and its various symmetries. ...
3
votes
3answers
85 views
What does $ds^2$ mean and how does it specify a metric?
Let $H$ be the upper half-plane in $\mathbb{R}^2$. How does the following expression $$ds^2= \frac{dx^2+dy^2}{y^2}$$ specify a Riemannian metric on $H$?
I don't understand what the expression means. ...
1
vote
1answer
81 views
Least distance on Riemannian Manifold
I've been doing some calculations of geodesics in different Riemannian Manifolds. More precisely I'm trying to compute, given two points on a Riemannian Manifold, the smallest distance between those ...
0
votes
1answer
100 views
A question about covariant derivative of a tensor
Let $R'$ be a tensor of order 4 in a riemannian manifold $M$ defined by:
$R'(W,Z,X,Y)=\langle W,X \rangle \langle Z,Y\rangle - \langle Z,X\rangle \langle W,Y\rangle $
And let $R$ be the curvature ...
2
votes
1answer
206 views
control of the $C^{1}$ norm of a diffeomorphism
Let $\mathcal{E}$ be the set of smooth manifolds with boundary $E\subset \mathbb{R}^{3}$ which are perturbations of the unit ball whose volume $V$, diameter $d$ and area of the boundary $A$ satisfy:
...
4
votes
2answers
188 views
An exercise in the Riemannian geometry book
If $M$ is a smooth closed $n$-dimensional Riemannian manifold which is Riemannian embedded in $\mathbb R^{n+1}$, then there exists a point $p \in M$ such that the sectional curvatures at $p$ are all ...
2
votes
1answer
42 views
Bounding the injectivity radius from below.
Let $(M, g)$ be a finite-dimensional Riemannian manifold, and let $S \subseteq M$ be a compact set.
I want to prove the following fact:
There exists a number $\epsilon > 0$ such that the ...
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
0answers
22 views
Diagonable Kernels over a Riemannian Surface
This question is motivated by this paper. There, they develop a stippling method which requires a kernel to be diagonal. Meaning a symmetric bilinear function
$K\colon \chi\times \chi\to \mathbb{R}$
...
7
votes
2answers
184 views
Any compact embedded $2$-dimensional hypersurface in $\mathbb R^3$ has a point of positive Gaussian curvature
Problem statement: Let $M \subseteq \mathbb{R}^3$ be a compact, embedded, 2-dimensional Riemannian submanifold. Show that $M$ cannot have $K \leq 0$ everywhere, where $K$ stands for the Gauss ...
3
votes
2answers
59 views
Critical paths for length cannot have kinks.
This problem is in Spivak's Differential Geometry (Ch.9 #37), and he gives a sketch of a proof which I have been unable to finish.
So let's compute $\frac{dL(\overline{\alpha}(u))}{du}\mid_{u=0}$ ...
11
votes
1answer
128 views
Dolbeault Cohomology is invariant under homeomorphisms
If $X$ and $Y$ are two complex manifolds, which are homeomorphic but not necessarily diffeomorphic, must their Dolbeault cohomology groups be isomorphic? Here the Dolbeault cohomology groups ...
2
votes
0answers
70 views
Connection between covariant derivative and basis vectors.
I read here, Curvilinear page 11, that $$\frac{\partial}{\partial x^i}e_j=\Gamma^k_{ij}e_k$$ where the $e_i$'s are basis vectors. There seems to be some connection, but when I calculate it, for ...
1
vote
0answers
69 views
What is the Weitzenböck formula for the $\bar\partial$-Laplacian?
It is well-known that the Weitzenböck formula for the real Laplacian is
$$
\Delta |\nabla f|^2 =|\operatorname{Hess} f|^2 + \langle \nabla f, \nabla \Delta f\rangle + \operatorname{Ricci}(\nabla f, ...
3
votes
0answers
57 views
Length of closed geodesics in a compact and simply connected manifold X
I have a Riemannian manifold $(X,g)$ which is compact, simply connected and with sectional curvature upper bounded by $k>0$ everywhere. Let $p\in X$ be any point and $q\in Cut(p)$ the nearest cut ...
1
vote
0answers
47 views
Solving to get free falling coordinate as function of arbitrary coordinate
From weinberg's gravitation, EQ : $3.2.11$
$$\frac{\partial^2 \zeta^\alpha}{\partial x^\mu \partial x^\nu} = \Gamma^\lambda _{\mu \nu}\frac{\partial\zeta^\alpha}{\partial x ^\lambda}$$
The solution ...
2
votes
0answers
107 views
Prerequisites for studying Hodge theory and the Hodge conjecture
To what branch of mathematics does the Hodge conjecture belong? I'm aware that it's very advanced, but what kind of prerequisites would one need to understand those problems? Can you suggest some good ...
3
votes
2answers
54 views
Invariance of curvature under a conformal mapping
Let $\Omega_{1}, \Omega_{2} \subseteq \mathbb{C}$ be bounded domains. Let $\rho$ be a metric on $\Omega_2$ and $h: \Omega_1 \rightarrow \Omega_2$ a conformal mapping. Let $$h^*\rho(z) = ...
5
votes
1answer
95 views
showing zero curvature implies a line
How can I show that a given (not necessarily unit-speed) parametrization $\gamma(t)$ of a curve in $\mathbb{R}^3$ which exhibits zero curvature is a line ? What I know is that zero curvature means ...
1
vote
1answer
62 views
curvature of space curve
I am slightly confused by the following curve $\gamma(t) = (e^t,0,0)$ in $\mathbb{R}^3$. Its curvature, defined as
$$
\kappa(t) = \frac{\|\dot \gamma(t) \times \ddot \gamma(t)\|}{\|\dot \gamma(t)\|^3}
...
12
votes
2answers
143 views
Computation of Laplace-Beltrami operator in a conformally equivalent metric
Could anyone tell me where I'm wrong with the following elementary calculation? Given a smooth Riemannian manifold $(M, g)$, I'd prove that if $\tilde{g}$ is conformally equivalent to $g$ (that is, ...
7
votes
4answers
263 views
Is length adimensional when space is not flat?
Consider the two manifolds $\mathbb{R}^2$, equipped with the usual metric $g_{ij}=\delta_{ij}$, and $\mathbb{H}^2=\{(x, y)\,:\,y>0\}$, equipped with the hyperbolic metric $h_{ij}=\delta_{ij}/y^2$. ...
