Milnor's Theorem

If a Compact Manifold has Nonnegative Ricci Curvature, then its Fundamental Group has at mostPolynomial growth. On the other hand, if has Negative curvature, then its Fundamental Group hasexponential growth in the sense that grows exponentially, where is (essentially) the number ofdifferent ``words'' of length which can be made in the Fundamental Group.

References

Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994.

• Exponential Inequality
• Exponential Integral
• Exponential Map
• Exponential Matrix
• Exponential Ramp
• Exponential Sum Formulas
• Exponential Sum Function
• Exponent Laws
• Exponent Vector
• Exradius
• Exsecant
• Extended Cycloid
• Extended Goldbach Conjecture
• Extended Greatest Common Divisor
• Extended Mean-Value Theorem
• Extended Riemann Hypothesis
• Extension
• Extension Problem
• Extensions Calculus
• Extent
• Exterior
• Exterior Algebra
• Exterior Angle Bisector
• Exterior Angle Theorem
• Exterior Derivative