Pressley's Elementary Differential Geometry requires little more than some multivariable calculus and linear algebra. It treats curves and surfaces in $\mathbb{R}^{3}$. However, the author is careful to point out that while many of the results generalise to higher dimensions, the methods used in the book do not always do so. This is done in part to make the subject accessible. It might be worth a look to get a taste of differential geometry without the machinery developed in more advanced courses on topology, smooth manifolds and the like.
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