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Would a torus shape be considered to have both positive and negative Gaussian curvature?

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Yes, there are many of those surfaces. In general saddle points will result in negative Gaussian curvature because the two principle radii of curvature are opposite in sign whereas peaks and holes will result in positive Gaussian curvature because their principle radii of curvature have the same sign (either both negative or both positive).

A few examples of surfaces with both positive and negative Gaussian curvature can be readily found in nature: a pear (usually), a peanut shell, a baseball bat (some negative Gaussian curvature at the handle).

A torus indeed has both positive and negative Gaussian curvature, because it has saddle points, a whole ring of them at the inside of the torus, and it has `peaks' at the outside. So it depends whether you are inside (in the hole) or outside the torus whether the Gaussian curvature is negative respectively positive (see e.g. this module on curvature).

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The product of k1 and k2 does not change if you are inside of the torus or outside it. –  Narasimham Aug 13 at 18:30
    
Could you clarify what you mean with k1 and k2?! --- And just to clarify my own wording: with inside I mean at the side of the torus inside the hole, I do not mean actually inside the 'tube' of the torus –  Michiel Aug 14 at 5:12

Consider a torus $T$ in $\Bbb R^{3}$. By the following question, $T$ has a point of positive curvature.

Any compact embedded $2$-dimensional hypersurface in $\mathbb R^3$ has a point of positive Gaussian curvature

By the Gauss-Bonet theorem, $\int_T K \ dA = \chi(T)=2-2g=0$, where $g=1$ is the genus of the torus. Thus, since $T$ has a point (and therefore a neighborhood) of positive curvature, it must have a neighborhood of negative curvature in order to make the above integral vanish.

This might be overkill, but I like it.

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It is a theorem of Hilbert that any closed smooth surface without boundary in 3 space must have a point of positive Gauss curvature. So any surface that also has a point of negative Gauss curvature will have both.

The integral of the Gauss curvature over a surface is 2pi times its Euler characteristic. The torus has Euler characteristic zero so it must always have regions of negative Gauss curvature to cancel the regions of positive curvature guaranteed in Hilbert's theorem. There is no way to warp it so that its Gauss curvature has only one sign. Similarly surfaces with more than one handle all have negative Euler characteristic and thus must always have both positive and negative Gauss curvature.

This is not true for the torus in 4 space. Here the torus can be given zero Gauss curvature everywhere.

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Such surfaces are called "anticlastic" surfaces. A "synclastic" surface has the center of curvature of any plane intersection always on the same side of the surface.

It can be proved, that for any continuous anticlastic surface, at any point on the surface, there is at least one plane intersection direction where the curvature is zero.

Well it has to go to zero to get from positive curvature to negative curvature, if it is continuous.

As a result, anticlastic surfaces are susceptible to buckling, along a direction of near zero curvature. That's why egg shells, are built synclastic, and not anticlastic.

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If a sine curve is rotated around an axis after being sufficiently displaced parallel to x-axis we have Gauss curvatures positive, zero and negative on the swept surface.

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If one has to perceive the Gaussian curvature in physical manner, consider the following figure.

a busy cat

The green dot denote the region having negative Gaussian curvature while the red dot denote the region of surface having positive Gaussian Curvature and the blue dot denote the region having zero Gaussian curvature(which means flat surface). Gaussian Curvature is the curvature of the surface.It can be resolved in two orthonormal directions i.e. results in forming two curves which lie on the surface cutting each other in the orthonormal direction at the points(shown in the figure with different colors). If the two orthonormal curves on the surface having their centres of curvature of the curves lying on the same side of the surface denotes Positive Gaussian curvature while the centres of curvature of the curves lying on different side of the surface denotes Negative Gaussian curvature. so, In Torus, when travelling from the regions near point of Green color to the regions near the point of Red color, the Gaussian Curvature varies from negative to positive as a result of which, there are points in between those have zero Gaussian Curvature.


http://en.wikipedia.org/wiki/Torus#mediaviewer/File:Torus_cycles.svg


Magnitude of the positive and negative curvatures depends on the radii of red and pink circles of the torus in the link above.

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