If net positive mass - energy means positive curvature, and similarly negative mass-energy means negative curvature, then spacetime with zero curvature implies zero mass energy.

General Relativity/Curvature

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The geometry taught in schools is Euclidean geometry; the geometry of a flat surface. Here all the familiar axioms apply, e.g. the angles of a triangle add up to 180^o, and the area of concentric circles increases proportionally to the square of the radius. However on a curved surface, e.g. the surface of a sphere, these axioms no longer apply. The angles of a triangle can add up to as much as 270^o, and flat-surface geometry no longer works. Such a surface is said to have a positive curvature.
Negatively curved surfaces also exist - they are shaped somewhat like an infinitely extended saddle - and Euclidean geometry does not apply to these surfaces either. For example, the angles of a triangle add up to less than 180^o.
If we extend these ideas to three dimensions, (do not be worried if you can't imagine a three-dimensional surface of a sphere, the human mind was never equipped to do so), we have three options to describe the geometry of the universe. Either:

1. The curvature of space is zero: i.e. Euclidean geometry applies
2. Space has positive curvature, i.e. it is shaped as a hypersphere(3D spherical surface)
3. Space has negative curvature, i.e. it is shaped like a so-called hypersaddle

Latest Limits on anisotropy of background radiation from WMAP
WMAP now places 50% tighter limits on the standard model of cosmology (cold dark matter and a cosmological constant in a flat universe), and there is no compelling sign of deviations from this model.
WMAP has detected a key signature of inflation. Wmap data place tight constraints on the hypothesized burst of growth in the first trillionth of a second of the universe, called 'inflation', when ripples in the very fabric of space may have been created. The 7-year data provide compelling evidence that the large-scale fluctuations are slightly more intense than the small-scales ones, a subtle prediction of many inflation models.
NASA's WMAP project showed to within 2% accuracy, by measuring angles between notable features in the Cosmic Microwave Background , that the universe is indeed flat (not in the pancake sense of the word, but meaning that it obeys the laws of Euclidean geometry). This has several intriguing implications (for example it implies that the total mass-energy of the universe is zero), some of which are covered later in this article.

General Relativity and Spacetime Curvature[edit]

Einstein's brilliance was to suggest that although gravity manifests itself as a force, it is in fact a result of the geometry of spacetime itself. He suggested that matter causes spacetime to curve positively. The sun, for instance warps spacetime, and it is this warping of geometry to which the planets react and not directly to the sun itself. This is a central tenet of the General theory of Relativity. This local curvature can be described in mathematical terms using tensor calculus, an incredibly elegant tool which provides consistent results, regardless of the chosen frame of reference.
This predicts that if a giant triangle was to be constructed around the sun, the angles at its vertices would in fact add up to more than 180^o. This is easy to imagine if one thinks of the sun as warping geometry, causing the triangle to have "wonky" sides. However it is incredibly important to note that these lines are in fact the straightest lines possible (geodesics) in this warped geometry.
These predictions can be tested, and have been to a very high degree of accuracy.

How come matter doesn't cause the universe to have an overall positive curvature?[edit]

How can the universe exhibit Euclidean geometry if stars and planets distort it locally? Einstein's mass - energy equivalence predicts that not only stars and planets contribute to this local distortion, but energy also does. So mass, the Cosmic Microwave Background and other electromagnetic energy all contribute to the positive curvature of spacetime. How, then, is the universe flat?
The answer lies in a curious fact about gravitation. Imagine if you were to pluck the Earth from its orbit, so that it was no longer affected by the gravitational field of the sun. You would have to expend energy to do so, implying that the potential energy possessed by the earth by virtue of its position in orbit around the sun is in fact, negative, as it requires an input of energy to raise it to a state of zero gravitational potential.
Now if mass and observed "positive" energy, cause spacetime to curve one way, then gravitational "negative" energy must curve it the other way, leading to the observed universe with zero curvature.
Consider this for a moment. If net positive mass - energy means positive curvature, and similarly negative mass-energy means negative curvature, then spacetime with zero curvature implies zero mass energy.