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'.
2026-03-30 17:37:01.1774892221
How to understand it will sweep out a 2-dim manifold?
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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.