I think the (a) generalisation you look for is totally geodesic sub manifold: a submanifold is totally geodesic if every geodesic of the submanifold is a geodesic of the ambient space. You can relax that to geodesics that are tangent to a given subbundle of the tangent bundle of the sub manifold.

@roberlow: yes they are more restrictive than the examples. That is why I suggested having a subbundle in the tangent bundle such that only geodesics with tangent in this subbundle need to be geodesics in the ambient space.

Incidentally in complex geometry there is a different notion of ruled varieties: a variety is ruled iff it is covered by rational curves (i.e. curves of genus 0). A smooth complex surface is ruled if and only if and only if the (or a) minimal model is CP^2 or a projectivised vectorbundle over a curve.

@Prof.Brussee @ Prof.Low. Thank you for the answers. I am sorry if I left the question a bit vague. I was looking at a generalisation of a ruled surface in a manifold, with the straight line replaced by the corresponding geodesic on the manifold.

I have heard of Totally geodesic manifolds , and I believe that every totally geodesic submanifold of a Riemannian Manifold is minimal.

Also totally umbilical manifolds which are minimal are totally geodesic as a weaker converse.

But I am not sure how that ties in with this.

Of course, a surface of revolution generated bu a geodesic would be quite similar to a "ruled surface" notion, wouldnt it?

@Vishsesh Bhat. If you have a surface swept out by lines that means you have an embedding S^1 \times R \to R^3 for some one dimensional curve S^1 (which is normally a circle hence the notation) such that for each \theta \in S^1 the restriction \theta \times R \to R^3 is linear i.e. is a geodesic. The tangent of this geodesic is contained in the subbundle TR \into T(S^1 \times R) \into TR^3|_{S^1 \times R}. That is why I think that the weakened version of totally geodesic where you only insist that tangents to some given subbundle are geodesics is a proper generalization of the notion of surface (or more generally sub manifold) swept out by lines. Lets call it partially geodesic for lack of a better name. The existence of a subbundle that admits any geodesics should be non trivial condition. Maybe a better definition of partly geodesic is that one should ask in addition that the subbundle is (locally) spanned by tangents of geodesics with tangent in the subbundle. This is certainly true for your ruled surfaces.

A totally geodesic foliation (of dimension 1) would be equivalent to the idea of a ruled surface. A ruled surface in space is indeed foliated by those lines (ignoring comes) which are geodesics on the surface.

In order to talk about a manifold being minimal, it needs to be a submanifold of some other space. It has no meaning except in that context.

There is some confusion on this thread between a totally geodesic (sub)manifold and a totally geodesic foliation of a manifold. Totally geodesic submanifold again are defined only in the context of being a submanifold of some other space, but a totally geodesic foliation (the proper generalization here) is a collection of lines in the manifold, just like a ruled surface.

A foliation is a decomposition of a manifold into submanifolds, called leaves, like the pages of a book or the lines of a ruled surface. To say that the foliation is totally geodesic is to say that each leaf is a totally geodesic submanifold of the total space, again like the lines in a ruled surface. Of course that ruled surface in addition sits in Euclidean space, and those lines are in addition geodesics in the ambient Euclidean space, but that adds an additional layer.

@David Johnson. You are right, I think that totally geodesic foliation is indeed the right notion here. However @Vishesh Bath seems to actually ask about the case of embedded surfaces, and then the "additional layer" seems to distinguish a ruled surface from say an embedded torus.

True. But, beyond the classical case, I don't think much has been done on that. That might actually be somewhat interesting.

@Prof.Johnson. I think your idea comes closer to what I was looking for. Nevertheless, to be honest, I think my question was a bit too vague. Thank you.

As Prof. Brussee said, I am indeed looking for embedded submanifolds.

Professor Bhat, this entire thread is very interesting, and it exceeds your original question. I think this paper/preprint may be somewhat helpful in regard to your original (first) question:

http://people.sju.edu/~ktapp/MT.pdf

ON TOTALLY GEODESIC FOLIATIONS AND DOUBLY RULED

SURFACES IN A COMPACT LIE GROUP

MARIUS MUNTEANU, KRISTOPHER TAPP

@ Prof.Toda, thank you for the link and apologies for the late acknowledgment. Also, I am nowhere near being a professor, merely a student and would be much obliged if you call me by my name.

In my opinion, the most general extension of "ruled surface" occurring as submanifolds N of a general Riemannian manifold M are "submanifolds which are submanifolds foliated by k-dimensional totally geodesic submanifolds of $M$, where k is a fixed integer satisfying 1≤k< dim N".