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I would like to get a geometric intuition behind convexity of hypersurfaces in Riemannian manifolds: Recall that a hypersurface in some Riemannian manifold is said to be convex, if its second fundamental form is positive definite. Can you tell me, what this means in terms of geometric pictures? To make it a bit more precise, what I mean by a geometric picture: For example a smooth function is convex, if its graph always lies below its secants. Is there any similar picture to keep in mind, when thinking about a hypersurface in a Riemannian manifold with positive definite secomnd fundamental form? Every help will be appreciated.

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Positive definite second fundamental form is morally equivalent to all second derivatives being positive (just like for a function from the real line to itself). But in Riemannian manifolds, you're taking these derivatives after manipulating the tangent plane to be on the $xy$-plane (or $x_1...x_{n-1}$ plane) centered at the origin. So this just means that locally, at each point the hypersurface bends away from the point like a sphere (or an ellipsoid), and never has a saddle point. So it really means the hypersurfaces behave locally like spheres or ellipsoids, and share many of the same properties (like convexity in the usual sense).

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