Matt
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Understanding iterated covariant derivatives to define Sobolev spaces on manifolds
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Let $(M,g)$ be a Riemannian manifold with associated Levi-Civita connection $\nabla$. Typically one introduces the covariant derivative on vector fields, i.e., $\nabla:\Gamma(TM)\times\Gamma(TM)\to\...

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Clarifying the definition of Lie derivative for tensors
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First of all, Lie derivatives are defined on any smooth manifold, so the Riemannian structure is superfluous. Let $T$ be a tensor field and $X$ a vector field. If you wish to look at the Lie ...

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Ricci Tensor, Curvature and Scalar Curvature computation from definition
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Using the definition of the Riemann curvature tensor $$R_{XY}Z=\nabla_{[X,Y]}Z-[\nabla_X,\nabla_Y]Z,$$ we can compute that $$R^a_{\;bcd}=\partial_d\Gamma^a_{cb}-\partial_c\Gamma^a_{db}+\Gamma^a_{de}\...

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How to prove Penrose "Bianchi symmetry" with non-zero torsion tensor using abstract indexing?
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3 votes

First, I think you have some notation backwards. We typically use parentheses for symmetrization and square brackets for antisymmetrization, i.e, $$T_{(ij)}=\frac{1}{2}(T_{ij}+T_{ji}),\qquad T_{[ij]}=...

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Proving completeness of subset in $\mathbb{R}^2$
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3 votes

Consider $$x_n=\left(0,\frac{\pi}{n}\right).$$

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Meaning of the supra- and sub-indexes in tensor notation
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2 votes

I'm not entirely sure I followed what your issue is, but it seems to be notational in nature, so perhaps the following exposition will be helpful. Let $V$ be a real linear space with associated dual $...

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Doubts about a definition of the variational derivative
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2 votes

I find this an interesting topic which is often ignored in certain Riemannian texts when introducing variational methods when dealing with energy and length minimization. Some of the rigorous ...

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Stack of records theorem from Allan Pollack and Guillemin differential topology.(Q.1.4.7)
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2 votes

By the Regular Level Set Theorem, since $y$ is a regular value of $f:X\to Y$, we have that $f^{-1}(y)$ is smooth embedded submanifold of $X$ of codimension (in $X$) equal to the dimension of $Y$. ...

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What are the laplacian operators for the three two dimensional metrics of one variable dependence?
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2 votes

I'll do (A). Recall, that in coordinates, we have that $$\Delta_g u=\frac{1}{\sqrt{|g|}}\partial_a\left[g^{ab}\sqrt{|g|}\partial_b[u]\right],$$ where $|g|=\det(g_{ab})$. Note for this case $g_{ab}=e^...

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Cogeodesic flow on compact manifold has compact leaves and manifold
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1 votes

This is equivalent to showing the sphere bundle $\pi:SM\to M$ is compact for compact manifolds $M$, and your case is just looking at the co-sphere bundle of radius $\sqrt{2\lambda}$, which are ...

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Definition of a left-invariant vector field
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1 votes

Let $G$ be a Lie group and $X$ a vector field. For $g\in G$, let $L_g:G\to G$ denote left-multiplication, i.e., $L_g(p)=gp$. Then we say that $X$ is left-invariant if $$(L_g)_*X=X.$$ Note that the ...

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Derivatives Across Summations
1 votes

I can see the confusion in the notation, so let's change things up a little bit to try and make it clearer. First let's note what our coordinates are: For convenience, all indices range from from $...

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Choosing Coordinate Charts -- Smooth Functions on Manifolds
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1 votes

Suppose $F^*(C^\infty(N))\subseteq C^\infty(M)$, that is, for any $g\in C^\infty(N)$, we have that $g\circ F\in C^\infty(M)$. Fix $p\in M$, and let $(\psi:W\subseteq N\to\tilde{W}\subseteq\mathbb{R}...

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Shortest path to a geodesic
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1 votes

It's possible I'm not understanding your question correctly, but let's think of this (more general) approach: Let $\alpha:I\to M$ be an curve on some Riemannian manifold $(M,g)$ (this could be your ...

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Tensor and Vector Notation
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1 votes

What you're looking for is the raising/lowering of indices, and the symmetrization/antisymmetrization of tensors. I'm not certain as to why you have the vector $V$, but I'm going to assume a follow ...

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Proof: When $f$ converges to $0,$ then $\frac 1f$ diverges.
0 votes

Suppose $f:\mathbb{R}\to\mathbb{R}$, $f(x)>0$ for all $x\in\mathbb{R}\setminus\{x_0\}$ and $$\lim_{x\to x_0}f(x)=0.$$ Let $N\in\mathbb{N}$, and let $\delta>0$ be such that $$|f(x)-0|<\frac{1}{...

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riemannienne geometry , differential calculus
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The manifold series of books by John Lee (Topological, Smooth, and Riemannian). Petersen's Riemannian geometry is a good one to go along with it as well. For the more advanced reading, Sakai or ...

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Derivative of determinant at some point
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For finding $d(\det)_A$, let's use what you've already computed. Given $B\in M_n(\mathbb{R})$, we have the curve $c(t)=A\exp{tA^{-1}B}$ with $c(0)=A$ and $c'(0)=B$. Thus \begin{align*} d(\det)_A(B)&...

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Even Functions, Symmetry, Inverse Functions
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The function is even, i.e., $f(-x)=f(x)$ (odd means that $f(-x)=-f(x)$ which yields rotational symmetry about the origin, which does not exist in this graph), which tells us that $b=d=0$ (the even ...

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Chain rule help? Property check
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The derivative of a linear map is itself. Consider $A:\mathbb{R}^n\to\mathbb{R}^n$, $x\mapsto Ax$, by matrix multiplication. Then the derivative of $A$ at the point $x$, is the unique linear map $...

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