The generalized Marcum Q-function, Q _m (a,b), is here explained geometrically as the probability of a 2m-dimensional, real, Gaussian random vector, whose mean vector has a Frobenius norm of a, lying outside a hypersphere of 2m dimensions, with radius b, and centered at the origin. Based on this new geometric interpretation, a new closed-form representation Q _m (a,b) is derived for the case where m is an odd multiple of 0.5. This representation involves only the exponential and the erfc functions, and thus is easy to handle, both numerically and analytically. For the case where m is an even multiple of 0.5, Q _m+0.5 (a,b) and Q _m-0.5 (a,b), which can be evaluated using our new representation mentioned above, are shown to be tight upper and lower bounds on Q _m (a,b), respectively. They are shown in most cases to be much tighter than the existing bounds in the literature, and are valid for the entire ranges of a and b concerned. Their average is also a good approximation to Q _m (a,b)