Talking about periodic signals does not require any knowledge of sines / Fourier analysis, nor reference to any physical systems. The no-nonense definition is just this:
A periodic signal with period π is a function π : β β β such that π : β+ is the unique smallest positive number for which
Β Β Β Β Β Β Β Β π(π‘ + π) = π(π‘) β π‘ββ
(there is no π < π that would also always give π(π‘ + π) = π(π‘)).
(We need the restriction forbidding π < π because else any signal with period π would also qualify as periodic with period 2Β·π or 3Β·π and so on.)
That readily allows us to also define the fundamental frequency as simply the reciprocal: πf = 1βπ.
As I promised, this does not yet make any reference whatsoever to sinusoidal components β in fact, it also applies to functions that cannot be represented by a Fourier series, such as
β§ 0 if π‘ is integral
π(π‘) := β¨
β© sin (1β1βcos(2Β·ΟΒ·π‘)) else

In this signal, you have in essence locally (close to 0, 1, 2 etc.) the ripple-frequency go up arbitrarily high, without decreasing its amplitude). This can't be constructed as just a superposition of constant-frequency sine parts. Thus it's also not really possible to talk of harmonics.
Now, such a βsignalβ fortunately can never actually turn up in the physical world β you'd basically have an ultraviolet catastrophe at each of the points where the frequency goes infinite. Actual physical signals, if they are periodic, can be decomposed into a superposition of sinusoidals, with the slowest-oscillating sharing the fundamental frequency of the signal. (That includes, of course, the audio example above, which is effectively bandlimited through Nyquist-sampling. Mind, this is for this signal not possible without hefty aliasing artifacts.)
And then, the answer to your question is yes, as topo morto explained: if a signal has no harmonics, then by definition of βharmonicβ, the signal is actually just a single sinusoidal.