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Suppose we have an Riemannian manifold $M$ endowed with Levi-Civita connection. Further, let $M_1$ and $M_2$ be two sub-manifold, which are isometrically embedded in $M$. We let $M_1$ roll over $M_2$ along a curve $\sigma_2:[0,T]\rightarrow M_2$. I found two notions of rolling map:

On this paper the rolling map should satisfy (Definition 2.1 on the paper):

Definition 1:

Rolling: Suppose $g:[0,T]\rightarrow \text{Iso}(M)$, then

  1. $g(t)\sigma_2(t)\in M_1$
  2. $T_{g(t)\sigma_2(t)}(M_1)=T_{g(t)\sigma_2(t)}(g(t)M_2)$

Further let $\sigma_1:[0,T]\rightarrow M_1$

No slipping

  1. $\sigma_1'(t)=g(t)_*\sigma'_2(t)$ ($g(t)_* $ is the push-forward)

No twist

  1. $(g'(t)g^{-1}(t))_*T_{\sigma_1(t)}M_1\subset T_{\sigma_1(t)}^\bot M_1$

  2. $(g'(t)g^{-1}(t))_*T^\bot_{\sigma_1(t)}M_1\subset T_{\sigma_1(t)}M_1$

Definition 2:

Alternatively, if we replace $M_2$ as the tangent space of $M$ at origin $T_oM_1$, the rolling map can be defined by parallel transport $\Gamma_0^s$ by along the curve $\sigma_1(t)$ satisfying $$\sigma_1'(s)=\Gamma_0^s(\sigma_1(s))\sigma'_2(s)\tag{2}$$

Question:

  1. It is clear that the definition of parallel transport and the condition (2) replaces the condition 3,4,5 in the first part. However, how is the condition 1,2 in the Definition 1 satisfied by Definition 2?
  2. Does the Definition 2 work if we replace the tangent space $T_oM_1$ with an arbitrary sub manifold $M_2$?
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