General question about dimensionality of the universe [duplicate]

I am a high school math teacher with some basic physic/lay person knowledge.

Each year after the AP test, I show the movies Flatland and Sphereland to my calculus classes and talk about the dimensions of our universe.

I know that there have been discussions of our universe having multiple dimensions other than the basic 3 with string theory and other theories. I believe that I also read that general consensus is that our universe tends more towards a Euclidean shape rather than an elliptical or hyperbolic type universe.

I also show a short Youtube video that is a hands-on exploration of gravity and how massive objects "warp" space-time to create gravity wells of some sort.

My question is this: If massive object really do warp space to create a gravity well model, is the universe warped in some 4th dimension that we don't notice similar to how the earth is essentially a 2d surface warped in the 3 dimension that we don't notice?

I don't treat what I tell them as the gospel truth, I just try to share what I've read and encourage them to study the ideas more if they are intrigued by the concepts.

Anything that could be shared so I can enlighten my students further would be great.

marked as duplicate by John Rennie general-relativity StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Oct 18 '18 at 15:14

What this comes down to is you are asking whether or not an $$N$$-dimensional manifold is necessarily embedded in an $$N+1$$-dimensional space. It's a fair question, since it seems like every time we humans are looking for intuition or draw pictures of, say, a sphere we embed it in 3D space.
It turns out that it is not the case that in order to sensibly talk about $$N$$-dimensional manifolds one must embed it in $$N+1$$ dimensions. I'm not going to prove it, as it's not what I would call a trivial fact, but it is nevertheless true. I recommend reading the "Manifolds" chapter of Sean Carrol's GR textbook. He has references and a more technical discussion if you wish to go down that road.