After reading a lot on induced connections, I am quite confused - I must admit I am a beginner lacking formal mathematics background. Essentially there appear to be two definitions which I am trying to reconcile. Let me attempt to clarify my problem:
Let $(\mathcal{S}, \boldsymbol{g})$ be a Riemannian manifold with connection $\boldsymbol{\nabla}$, and let $\mathcal{R}$ be immersed in $\mathcal{S}$ by the diffeomorphism $\phi : \mathcal{R} \rightarrow \mathcal{S}$. Then, the immersion induces the pullback-metric $\boldsymbol{g}^\phi = \phi^*\boldsymbol{g}$ on $\mathcal{R}$. From what I understand, the standard definition of an induced connection $\boldsymbol{\nabla}^\phi$ on $\mathcal{R}$ is essentially a consequence of the induced metric (see Lerman, E. Connections and Curvature Notes, p. 11), i.e. locally, the induced connection coefficients are simply built using $\boldsymbol{g}^\phi$ instead of $\boldsymbol{g}$. In this context, for a one-form $\boldsymbol{s} \in \Gamma(T^*\mathcal{S})$ and a vector-field $\boldsymbol{\nu} \in \Gamma(T\mathcal{R})$, I assume the induced connection would satisfy: $$\boldsymbol{\nabla}^\phi_{\boldsymbol{\nu}}(\phi^*\boldsymbol{s}) = \phi^*(\boldsymbol{\nabla}_{\phi_*\boldsymbol{\nu}} (\boldsymbol{s})) \quad \in \Gamma(T^*\mathcal{R})\,.$$ This appears to be quite common and also appears to be the Wikipedia definition, except that they use the tangent map instead of the push-forward.
A different definition on vector fields as elements of sections of pullback-bundles (Aubram, D. Differential Geometry Applied to Continuum Mechanics, Theorem 3.6.13, p. 47, and Bishop, R.L. and Goldberg, S.I., Tensor Analysis on Manifolds, eq. 5.7.1, p. 222), which I don't completely understand but which I require to treat two-point-tensors (with the same definitions as above):
A regular map $\phi$ induces a unique connection $\boldsymbol{\nabla}^\star$ on $\mathcal{R}$ such that for $\boldsymbol{t} \in T_P\mathcal{R}$, with $P \in \mathcal{R}$, and $\boldsymbol{v} \in \Gamma(T\mathcal{S})$ one has $$\boldsymbol{\nabla}^\star_{\boldsymbol{t}}(\boldsymbol{v} \circ \phi) = \boldsymbol{\nabla}_{T\phi(\boldsymbol{t})}\boldsymbol{v} \quad \in \Gamma(\phi^*T\mathcal{S})\, ,$$ where $(\boldsymbol{v} \circ \phi): \mathcal{R} \rightarrow T\mathcal{S}$.
Here, $T\phi: T\mathcal{R} \rightarrow T\mathcal{S}$ is the tangent map and $\Gamma(\phi^*T\mathcal{S})$ is a section of the pullback-bundle, such that $(\boldsymbol{v}\circ\phi) \in \Gamma(\phi^*T\mathcal{S})$ has legs in $\mathcal{S}$ but points in $\mathcal{R}$.
My questions now are the following:
- How are both definitions related?
- In the second definition, why is the right-hand-side in $\Gamma(\phi^*T\mathcal{S})$ and not in $\Gamma(T\mathcal{S})$ as would be expected from the usual covariant derivative $\boldsymbol{\nabla} : \Gamma(T\mathcal{S}) \times \Gamma(T\mathcal{S}) \rightarrow \Gamma(T\mathcal{S})$?
- To me, the "proof" of the second definition is confusing since point arguments have been omitted. Is there perhaps some insight that more knowledgable users can add?