1
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1answer
34 views

formula of square of the covariant derivative

I am stuck with the calculation of $(\nabla ^2 \beta)(X,Y,Z_1,\dots,Z_r)$. In the following, capital letters are arbitrary vector fields. Suppose $\beta$ is an $(r,0)$ tensor. Denote $(\nabla ...
4
votes
1answer
66 views

Classical tensor analysis and Tensors on Manifolds

I learned tensors the bad way (Cartesian first, then curvilinear coordinate systems assuming a Euclidean background) and realize that I am in very bad shape trying to (finally) learn tensors on ...
1
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1answer
62 views

Question Symmetric 2-Tensors on Vector Fields

I have a question on a direct computation. How would one compute the following $$ (dx \odot dy ) \left(\frac{1+uv+xy}{1+xy} \frac{\partial}{\partial v}, \frac{\partial}{\partial y} \right) $$ and$$ ...
2
votes
1answer
69 views

Construct tensors from differential forms?

Let $(M,g)$ be a Riemannian manifold, differential forms are defined using tensors, could we define a tensor using a differential form? For example, if $\omega$ is a two-form on $M$ which is expressed ...
0
votes
1answer
25 views

Why this doesn't transform properly?

We are in $\mathbb R^n $, with a tensor field of components $T_\nu$, and being $e_\mu$ the vectors of the basis: $e_\mu \equiv \partial_\mu$, then I'm asked to show that $\partial_\mu T_\nu$ can't ...
0
votes
1answer
21 views

Linear transformation from endormorphism to real number

For a finite dimensional vector space $V$, is there a linear transformation between its endomorphism and real number, please? I suspect that since the element of the endomorphism can be represented by ...
4
votes
1answer
112 views

Tensor notation (practicing)

I'm praticing tensor notation, and I want to prove this way that given vectors $A,B,C,D$ then $(A \times B) \times (C \times D) = \det(A,C,D)B - \det(B,C,D)A$, where $\det$ means the triple product. ...
2
votes
1answer
53 views

The relation between principal curvature and curvature tensor?

To me, there are two systems of curvature of a surface, one is consist of 'principal curvature, mean curvature, Guass curvature, normal curvature' while the other is consist of 'curvature tensor'. I ...
3
votes
1answer
99 views

The Curvature Tensor

I present three different ways I've seen the Riemann curvature written: $R(X,Y)Z=D_XD_YZ-D_YD_XZ-D_{[X,Y]}Z$ $R(e_c,e_d)e_b=D_{e_c}D_{e_d}e_b-D_{e_d}D_{e_c}e_b-D_{[e_c,e_d]}e_b$. $R^{\rho}_{\space ...
6
votes
1answer
79 views

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 $T_s^r (M)$ ...
2
votes
0answers
71 views

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 ...
2
votes
0answers
90 views

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 ...
1
vote
0answers
50 views

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 ...
3
votes
2answers
77 views

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 ...
2
votes
1answer
48 views

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 ...
0
votes
1answer
204 views

Christoffel Symbols and the change of transformation law.

I have seen it written that the change of co-ordinate form is given by the following: $$ \tilde \Gamma^{i}_{jk} = {\partial \tilde x^i \over \partial x^\alpha} \left [ \Gamma^\alpha_{\beta ...
1
vote
1answer
56 views

Metric tensors. Have I got the correct understanding?

My course is covering metric tensors in a slap-dash way, so I want to ensure I have understood correctly how they are described. I hope you can confirm this! So, I believe that a metric tensor is a ...
0
votes
1answer
79 views

Derivation for affine connection formulas on differentiable manifolds (General tensors)

Let $p\in U\subseteq M$ be a point in some neighborhood of a finite-dimensional differentiable manifold, $\{x^i\}$ a set of local coordinates with respect to $U$, and ...
1
vote
1answer
248 views

Riemann tensor in terms of the metric tensor

The Riemann tensor is a function of the metric tensor when the Levi-Civita connection is used. Most tensor notation based texts give the Riemann tensor in terms of the Christoffel symbols, which are ...
3
votes
1answer
90 views

Why the space of skew-symmetric tensors $\Lambda^{n}V$ is a one dimensional if $dim(V)=n$

While reading Liviu Nicolaescu Lectures on the geometry of manifolds, I came accross the notion of "determinant line": Definition: Lev $V$ be an n-dimensional R-vector space. The one dimensional ...
0
votes
1answer
28 views

choosing indices in tensor notation

I have the following operator, where $\rho$ is a scalar and $u$ is a vector: $$ \nabla (\rho u) - (\nabla \rho)u - u(\nabla \rho) $$ My book writes this in index notation as $$ \partial_\alpha(\rho ...
6
votes
1answer
299 views

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 ...
1
vote
1answer
59 views

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 ...
1
vote
2answers
68 views

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 $\nabla _a\nabla ^af=\nabla ^c\epsilon _{cb}\nabla ^bh$ ...
4
votes
1answer
101 views

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 ...
2
votes
1answer
62 views

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?
2
votes
1answer
132 views

How to translate between differential forms and tensor index notation

The books on Manifold theory & geometry that I studied introduce connection and curvature in the language of differential forms. But Physics books on the other hand like (General Relativity by ...
3
votes
1answer
128 views

A linear connection induces a covariant derivative of tensor fields.

Let $M$ be a smooth manifold. notation: $\mathcal T(M)^{(k,l)}$ is the $C^{\infty}(M)$-module of all tensor fields of type $(k,l)$ on $M$ ($k$ indicates the covariant part). $\mathcal ...
3
votes
1answer
75 views

Non commutative tensor products / simple example?

In O'Neil's "Semi-Riemannian Geometry" it is stated that if $A$ is a $(r,0)$-tensor field and $B$ is $(0,s)$-tensor field then it is ""from the definition'' that $A \otimes B = B \otimes A$. I still ...
4
votes
1answer
90 views

O'Neil's problem 2.10 on Tensor Derivations. Literature on the subject.

I am having real trouble trying to understand a problem in O'Neil's "Semi-Riemannian Geometry" and I can't find much literature on the subject. I will expose the problem and I will be grateful to ...
3
votes
1answer
406 views

What is the practical difference between abstract index notation and “ordinary” index notation

I understand that in "normal" index notation the indexes can be thought of as coordinates of scalar values inside a tabular data structure, while in the abstract index notation they can not. However, ...
2
votes
0answers
74 views

Tensors: summing over indices

Would anybody mind teaching me how to work these indices? Definitions: Throughout the following, repeated indices are to be summed over. Hodge dual of a p-form $X$: $$(*X)_{a_1...a_{n-p}}\equiv ...
9
votes
2answers
741 views

Do I understand metric tensor correctly?

So I've been studying vectors and tensors, and I'm trying to understand metric tensors. As I understand them, besides a vast array of explanations, they provide an invariant distance between vectors ...
4
votes
2answers
145 views

For covariant tensors, why is it $\bigwedge^k(V)$, not $\bigwedge^k(V^*)$?

In learning the very basics of differential geometry, I have seen the exterior product defined a couple of ways: First, I have seen it as the image of the covariant tensors (which I believe are ...
6
votes
2answers
369 views

Special case of the Hodge decomposition theorem

I am trying to prove the following special case of the Hodge decomposition theorem in differential geometry for an $n$ component vector field $V_i$ in $\mathbb{R}^n$. I have very little knowledge of ...
4
votes
1answer
246 views

Tensors as mutlilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely $$ V\otimes W := L_2(V^*\times W^*,\Bbb F) $$ I am also aware that this space is isomorphic to the tensor ...
3
votes
1answer
456 views

Basis of vector fields on manifold

For a typical real manifold of $n$ dimensions, the basis of a tangent vector space $T$ at point $p$ is $$\frac{\partial}{\partial x_i},\ldots,\frac{\partial}{\partial x_j}$$ So is the general basis of ...
2
votes
1answer
634 views

What is the metric tensor on the n-sphere (hypersphere)?

I am considering the unit sphere (but an extension to one of radius $r$ would be appreciated) centered at the origin. Any coordinate system will do, though the standard angular one (with 1 radial and ...
1
vote
1answer
34 views

Metric spaces and curvature

Suppose that it is given that the Riemann curvature tensor in a metric space of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a vector in the ...
1
vote
0answers
69 views

What is the needed background to study tensors?

What is the needed background to study tensors? I do not plan to study it but, I want to know the background! It's a matter of curiosity . Thanks.
2
votes
1answer
127 views

Prove that a tensor field is of type (1,2)

Let $J\in\operatorname{end}(TM)=\Gamma(TM\otimes T^*M)$ with $J^2=-\operatorname{id}$ and for $X,Y\in TM$, let $$N(X,Y):=[JX,JY]-J\big([JX,Y]-[X,JY]\big)-[X,Y].$$ Prove that $N$ is a tensor field of ...
2
votes
1answer
375 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 ...
3
votes
1answer
247 views

Parallel Transport along a curve

We had this homework assignment for our geometry course, and we couldn't figure it out, any ideas on how to do this: Consider the Poincare model of Lobachevsky plane, $H^2=\left\lbrace{ ...
1
vote
0answers
178 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 ; ...
2
votes
1answer
47 views

Regarding the definition of covariant derivative and its use on basis vector fields

we find that for general vector fields ${\mathbf v}= v^ie_i$ and ${\mathbf u}= u^je_j$ we get :$\nabla_{\mathbf v} {\mathbf u} = \nabla_{v^i {\mathbf e}_i} u^j {\mathbf e}_j = v^i \nabla_{{\mathbf ...
0
votes
3answers
218 views

How to differentiate a differential form?

Please explain me the idea of differentiating differential forms (tensors). Example: compute d(xdy + ydx) The answer is known, we should have 0. What's the rule?
1
vote
0answers
309 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 ...
3
votes
3answers
70 views

Definition of a tensor field

Could anybody explain to me the following: If $$T_{ij}=\nabla_i V_j-\nabla_j V_i$$ where $V_i$ is a covector field and $\nabla_i$ is the covariant derivative, then $T_{ij}$ is a tensor field. ...
5
votes
1answer
208 views

A tensor calculus problem

If the relation $a_{ij}u^iu^j=0$ holds for all vectors $u^i$ such that $u^i\lambda_i=0$ where $\lambda_i$ is a given covariant vector, show that $$a_{ij}+a_{ji}=\lambda_iv_j+\lambda_j v_i$$ where ...
5
votes
3answers
553 views

Definition of a tensor for a manifold

While reading Nakahara's geometry, topology and physics. I came across the following definition of a tensor. A tensor $T$ of type $(p, q)$ is a multilinear map that maps $p$ dual vectors and $q$ ...