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We use $(4+1)$ split to derive the 4D induced energy density $\rho $ andpressure $p$ of the $5D$ Ricci-flat cosmological solutions which arecharacterized by having a bounce instead of a bang. The solutions contain twoarbitrary functions of time $t$ and, therefore, are mathematically rich ingiving various cosmological models. By using four known energy conditions(null, weak, strong, and dominant) to pick out and study physically meaningfulsolutions, we find that the 4D part of the $5D$ solutions asymptoticallyapproach to the standard 4D FRW models and the expansion of the universe isdecelerating for normal induced matter for which all the four energy conditionsare satisfied. We also find that quintessence might be normal or abnormal,depending on the parameter $w$ of the equation of state. If $-1\leqslantw<-1/3$, the expansion of the universe is accelerating and the quintessence isabnormal because the strong energy condition is violated while other three aresatisfied. For phantom, all the four energy conditions are violated. Before thebounce all the four energy conditions are violated, implying that the cosmicmatter before the bounce could be explained as a phantom which has a largenegative pressure and makes the universe bouncing. In the early times after thebounce, the dominant energy condition is violated while the other three aresatisfied, and so the cosmic matter could be explained as a super-luminalacoustic matter.Comment: 10 pages. To appear in IJMP

USING (4+1) SPLIT AND ENERGY CONDITIONS TO STUDY THE INDUCED MATTER IN 5D RICCI-FLAT COSMOLOGY