Connections

Proposition 10.4. Suppose $\Phi:(M, g) \rightarrow(\widetilde{M}, \widetilde{g})$ is an isometry, then we have

$$ \Phi_*\left(\nabla_X Y\right)=\widetilde{\nabla}_{\Phi_* X}\left(\Phi_* Y\right) > $$

for any $X, Y \in \Gamma^{\infty}(T M)$. Here $\nabla$ and $\widetilde{\nabla}$ are the Levi-Civita connections for $g$ and $\widetilde{g}$ respectively. That is, the Levi-Civita connection is isometric-invariant

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connection satisfies $\nabla(f \sigma)=d f \otimes \sigma+f \nabla \sigma$ and $d\left\langle\sigma_2, \sigma_2\right\rangle=\left\langle\nabla \sigma_1, \sigma_2\right\rangle+\left\langle\sigma_1, \nabla \sigma_2\right\rangle$, where $f: M \rightarrow R$, and $\langle$, $\rangle$ is the Riemannian metric on $E=L_e \oplus \varepsilon$ which was described above.

$$ \Gamma=d x^\mu \otimes\left(\partial_\mu+\Gamma_\mu^i\left(x^\nu, y^j\right) \partial_i\right) $$ $$ \nabla(s)=\sum_{\ell=1}^n d x^{\ell} \otimes \nabla_{\ell}(s) $$ $$ \nabla_{\ell}(s):=\sum_{i, j=1}^k\left(\frac{\partial s^j}{\partial x^{\ell}}+\Gamma_{\ell i}^j s^i\right) e_j $$

Formulae

Connections

Let $\mathcal{M}$ be a smooth manifold and let $\Gamma(\mathrm{T} \mathcal{M}$ be the space of vector fields on $\mathcal{M}$, that is, the space of smooth sections of the tangent bundle $\mathrm{T} \mathcal{M}$. Then an affine connection on $\mathcal{M}$ is a bilinear map

$$ \begin{aligned} \Gamma(\mathrm{T} \mathcal{M}) \times \Gamma(\mathrm{T} \mathcal{M}) & \rightarrow \Gamma(\mathrm{T} \mathcal{M}) \\ (X, Y) & \mapsto \nabla_X Y, \end{aligned} $$

such that for all $f$ in the set of smooth functions on $\mathcal{M}$, written $C^{\infty}(\mathcal{M}, \mathbb{R})$, and all vector fields $X, Y$ on $\mathcal{M}$ :

1. $\nabla_{f X} Y=f \nabla_X Y$, that is, $\nabla$ is $C^{\infty}(\mathcal{M}, \mathbb{R})$-linear in the first variable;
2. $\nabla_X(f Y)=\partial_X f Y+f \nabla_X Y$, where $\partial_X$ denotes the directional derivative; that is, $\nabla$ satisfies Leibniz rule in the second variable.

Christoffel Symbols

Equivalent definitions:

• $\Gamma^A_{BC}(X) \mathbf{E}_A(X) = (\nabla_{\mathbf{E}_B}\mathbf{E}_C)(X)$
• $\nabla_j \partial_k=\Gamma_{j k}^l \partial_l$
• $\nabla_i \mathrm{e}_j=\Gamma_{i j}^k \mathrm{e}_k$

Lie Connection (The covariant derivative)

• A connection is Torsion Free iff (equivalently)

  • $$\nabla_i \partial_j-\nabla_j \partial_i=\left(\Gamma_{j k}^l-\Gamma_{k j}^l\right) \partial_l=\left[\partial_i, \partial_j\right]=0$$
  • $$\nabla_X Y-\nabla_Y X=[X, Y] \quad \forall X,Y$$

    where $X$ and $Y$ are vector fields.

  • $\Gamma^A_{BC} = \Gamma^A_{CB}$

  • The inner product (defined using $g$) between tangent vectors are preserved

• Metric compatibility/preservation; Leibnitz rule

  $$\partial_i\left(g\left(\partial_j, \partial_k\right)\right)=g\left(\nabla_i \partial_j, \partial_k\right)+g\left(\partial_j, \nabla_i \partial_k\right)$$

  or

  $$\nabla g=0$$

For the Levi-Civita connection,

$$ \Gamma_{k l}^i=\frac{1}{2} g^{i m}\left(\frac{\partial g_{m k}}{\partial x^l}+\frac{\partial g_{m l}}{\partial x^k}-\frac{\partial g_{k l}}{\partial x^m}\right) $$