# Tagged Questions

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### Einstein Summation - does the following equality hold: $a_{ij} x_i y_j = a_{ij} y_i x_j$

Does equality hold when $x_i = y_i$ and $x_j=y_j,$ and $i, j = 1, ..., n$.
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### Einstein Summation: How do I show $a_{ij} (x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j$?

Einstein Summation: How do I show $a_{ij} (x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j$?
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### normal vectors in spaces where $n > 3$

I am reading Lovelock and Rund's book on Tensors and they make a statement that I wanted to validate about normal vectors in high-dimensional spaces. It should be remarked that the above ...
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### Tensor fields and vector bundles

Let $M$ be a differentiable manifold, $TM$ and $T^*M$ a tangent and cotangent bundle of $M$ and let $\Gamma (TM),\ \Gamma (T^*M)$ be spaces of smooth sections of $TM$ and $T^*M$. Let $T_s^r (M)$ ...
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### Covariant vectors and Dual spaces

There are contravariant vectors and their duals called covariant vectors. But the duals are defined only once the contravectors are set up because they are the maps from the space of contravariant ...
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### Prove the Curvature Tensor is a Tensor

For an affine connection $\nabla$, prove the curvature R $R(X,Y,Z,\alpha)=\alpha(\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z -\nabla_{[X,Y]}Z)$ with $X,Y,Z$ vector fields and $\alpha$ a co-vector, is ...
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### Curvature tensor on the sphere

While reading R. Hamilton paper "Three-manifolds with positive Ricci curvature" I came across the sentence: For the sphere we have $$R(u,v,u,v)=R_{ijkl}u^{i}v^{j}u^{k}v^{l}>0,$$ which is the ...
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### What is the value of the 2-form $X(u(Y))-Y(u(X))-u([X,Y])$, is it $2du(X,Y)$?

Let $X,Y$ be two vector fields on a Riemannian manifold $(M,g,\nabla)$ where $\nabla$ is the symmetric metric connection on $M$ and let $u$ be a 1-form. I want to find the value or a simple form or ...
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### Why is it true that ${(\nabla Y)'^i}_j = {Y'^i}_{,j} + {\Gamma'^i}_{jk}Y'^k$?

I do not understand why this equation transforms as it does : ${(\nabla Y)'^i}_j = {Y'^i}_{,j} + {\Gamma'^i}_{jk}Y'^k$ Could someone give me a detailed explanation of why this is true please? I ...
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### Vectors, Basis, Dual Vectors, Dual Basis and Tensors

I'm trying to understand tensors and I know they have something to do with the basis and the dual basis of a vector space and a dual space. First I will give a concrete example to make clear what I ...
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### Why is this combination of a covariant derivative and vector field a (1,1)-tensor?

I have a question regarding something Penrose says in section 14.3 of The Road to Reality. It says '...when $\nabla$ acts on a vector field $\xi$, the resulting quantity $\nabla \xi$ is a ...
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### Covariant derivative in abstract index notation

Spose $f,h$ functions, where $\nabla _af = \epsilon _{ab}\nabla ^bh$. Then $\nabla ^af=g^{ac}\epsilon _{cb}\nabla ^bh$. My question is then does $\nabla _a\nabla ^af=\nabla ^c\epsilon _{cb}\nabla ^bh$ ...
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### Is invariance of a multi-linear form required for co/contra variance?

I'm reading the book: The Absolute Differential Calculus by Levi-Civita to get an idea of the history behind the development of tensor calculus. On page 71 he states: An m-fold covariant is an ...
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### Definition of divergence of a tensor

How do you formally define the divergence of an arbitrary $(p,q)$ tensor? And what does it geometrically signify?