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As red line in picture below, I really can't imagine why it will sweep out a 2-dim submanifold.

Below picture is from the 9th page of John M. Lee's 'Riemannian Manifolds: An Introduction to Curvature'.

enter image description here

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  • $\begingroup$ If the subspace $\Pi$ is $d$-dimensional, the geodesics whose initial tangent vectors belong to $\Pi$ must form a $d$-dimensional submanifold. Just imagine the neighborhood of the point $p$. $\endgroup$
    – Marcel
    Jan 16, 2016 at 14:12

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Author here: It's not surprising that you don't understand why -- this chapter is just supposed to suggest ideas that will be developed fully later in the book. This particular fact follows from properties of the exponential map, developed in Chapter 5. See also the bottom of page 145.

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  • $\begingroup$ Thanks very much. Your book is very good. $\endgroup$
    – Enhao Lan
    Jan 17, 2016 at 1:48

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