I don't know how to solve the following problem and would appreciate some help.

Let $M$ be a submanifold of euclidean space and $c:[a,b] \to M$ a non-constant curve, such that the velocity field of $c$ is a multiple of its acceleration field, i.e. $\frac{\mathrm{d}c}{\mathrm{d}t}= \lambda\frac{\mathrm{d}^2c}{\mathrm{d}t^2}$ $\forall t\in [a,b]$ and a $\lambda \in \mathbb{R}$. Now I have to show that $c$ is immersed, i.e. $\frac{\mathrm{d}c}{\mathrm{d}t}\neq0$ $\forall t\in [a,b]$ and that a constant speed reparametrization of $c$ is a geodesic.

For the second part I think the solution is as follows: Let $\psi$ be the diffeomorphism, such that $(c\circ\psi)(s)$ has constant speed. Then $\frac{\mathrm{d}(c\circ\psi)}{\mathrm{d}s}=K(=constant)$ and thus also $\nabla_{\frac{\mathrm{d}}{\mathrm{d}s}}\frac{\mathrm{d}(c\circ\psi)}{\mathrm{d}s}=\nabla_{\frac{\mathrm{d}}{\mathrm{d}s}}K=0$ (Here $\nabla$ denotes covariant differentiation). Hence $c\circ\psi$ is a geodesic. But I am not sure if this is right, and I don't know how to show the first part.


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