# Linear connection on a manifold: Math vs. Physics

I have learned some Riemannian Geometry in a strongly mathematical framework, precisely from the book "J.M.Lee - Riemannian Manifolds: An introduction to Curvature". Now I'm trying to learn Relativity from the Wald's book, but I have many problems to match the Riemannian Geometry notions from the mathematical framework to the physical one.

Consider the notion of linear connection: For me a linear connection $\nabla$ is a function $$\nabla:\mathcal T(M)\times\mathcal T(M)\longrightarrow\mathcal T(M)$$ $$(X,Y)\longmapsto \nabla_XY$$ where $\mathcal T(M)$ is the $C^\infty(M)$-module of sooth vector fields (sections of the tangent bundle). This function $\nabla$ has certain properties that allow to write $\nabla_XY$ in local coordinates. I know that exists an essentially unique way to define a (Koszul) connection $\overline\nabla$ on an tensor field starting from $\nabla$ and with the connection $\overline\nabla$ I can define a total covariant derivative for tensor fields. All these reasonings are done without computations in coordinates, but using strongly the "Tensor Characterization Lemma".

Wald instead says that a covariant derivative is a way to associate to a tensor field $T\in\mathcal T^{(k,\ell)}(M)$ another tensor field $\nabla T\in T^{(k,\ell+1)}(M)$ written in the index notation as $$\nabla_c{T^{a_1,\ldots,a_k}}_{b_1,\ldots,b_\ell}$$ such that it satisfies certain conditions. Now even if I understand the abstract index notation (infact I recognize that $\nabla_c{T^{a_1,\ldots,a_k}}_{b_1,\ldots,b_\ell}$ is a $(k,\ell+1)$-tensor) I don't understand how this approach coincides with the above one.

Thanks in advance.

• Just a side comment on books -- you might enjoy O'Neill's book "Semi-Riemannian Geometry and Applications to Relativity." – Neal Nov 9 '13 at 12:58
• I too am reading Wald's book after studying Manifold theory from the book by Loring Tu. There is a different but perhaps somewhat related question that I asked here that you may like to see. I too am interested in knowing some good references on Tensor algebra & calculus that will make me more comfortable translating between the two notations more efficiently. Thanks for asking this question. – user90041 Nov 9 '13 at 13:37

## 1 Answer

Your first definition of $\nabla$ often is referred to as covariant derivative of a vector field. More general, you can define a connection $\nabla$ on a vector bundle $E$ as a map between sections on $E$ $$\nabla:\Gamma(E)\to\Gamma(E\otimes T^*M)\qquad\text{satisfying}\qquad\nabla(fe_1+e_2)=e_1\otimes\mathrm{d}f+f\nabla e_1+\nabla e_2$$ for $e_i\in\Gamma(E)$ and $f\in C^\infty(M)$. This is Wald's approach.

On the other hand, if you "insert" a vector field $X\in\mathcal{T}(M)$ into the second component, you obtain a map $$\nabla:\Gamma(E)\times\mathcal{T}(M)\to\Gamma(E)$$ which is commonly denoted as $\nabla e(X)\equiv\nabla_X e$ and in the case of $\Gamma(E)=\mathcal{T}(M)$ is exactly how Lee introduces a linear connection.

However, I am not sure, if the two notions can be summarized as "Math vs. Physics" since both notions are used in mathematics (as far as I know).

• Indeed, both notions (and others besides, e.g., Ehresmann's definition) are constantly used within differential geometry alone—all are equivalent to each other, though some are more useful in certain contexts or for certain computations than in others. – Branimir Ćaćić Nov 10 '13 at 5:01