The theme of rivalry in The Rivals is one of the central themes since it is the titular theme, though the theme of Romantic, irrational sentimentality is the most directly pointed theme.
LYDIA ... we had never had a quarrel, ... I was afraid he would never give me an opportunity [to]. So, last Thursday, I wrote a letter ... [and] charged him with ... falsehood, put myself in a violent passion, and vowed I'd never see him more. [...] I intended only to have teased him three days and a half, ....
Yet, the theme of the title is without doubt important as there is at least one direct rivalry and at least one illusionary rivalry. Absolute, by a clever plot device, is in a rivalry with himself and at the same time with Acres, Julia's near neighbor. All, the Ensign, Absolute, and Acres are vying for Lydia's affection and hand in marriage. Absolute is in a rivalry with himself because he has adopted a false identity to make himself acceptable to Lydia who has wild Romantic sentimentalist notions of forfeiting her inheritance and running off to elope with an impoverished gentleman, thus bringing her name to shame and closing every social door to her and her husband. Absolute has feigned the identity of a lowly ensign when he is in fact an heir. He is in a standard sort of rivalry with Acres who loves Lydia himself and, of course, wants to outdo Absolute.
Of the illusionary rivalry, it might be said that Faulkland is in--or rather fancies himself in--a rivalry with Lucy's emotional state. When he is away from her, he is in an utter torment of unhappiness for himself and for what he imagines to be her forlorn and sad condition. It is Acres' information that Lucy has been cheerful, danced, sung, and the light of every social gathering, that undoes Faulkland's happiness even further. Now he cannot stop thinking that she must be very inconstant in her love to be able to sing and dance without him--and such a song!:
ACRES ... now I recollect one of them—My heart's my own, my will is free.
This illusionary rivalry between Faulkland and Julia's emotions results in contests to prove her sincerity. Lucy eventually learns to loath his continual distrust and very rationally sends Faulkland away for good. It is only the intervention of Sir Anthony's recommendations on Faulkland's behalf that reunites them and preserves their wedding day.
JULIA ... you have trifled with my sincerity ——[...] I now see it is not in your nature to be content or confident in love. With this conviction—I never will be yours. [...] it lost you the love of one who would have followed you in beggary through the world!
FAULKLAND She's gone—for ever!,
Sir ANTHONY ... Come, Julia, ... All the faults I have ever seen in my friend Faulkland ... proceed from ... his affection for you—There, marry him directly, Julia; you'll find he'll mend surprisingly!
In this satirical comedy of manners, the theme of rivalry is meant to illustrate the ludicrousness of heightened sentimentality (Lydia & Faulkland) and foolishness (Acres) while applauding rational and reasonable behavior (Julia & Absolute); though Absolute uses trickery, he is acting and thinking reasonably while finding a way past Lydia's prejudices and sentimentality. The theme of rivalry is bound up in the theme of sentimentality:
Look on her well—
Do solemn sentiments become that mouth?
Bid her be grave, those lips should rebel prove
To every theme that slanders mirth or love. (Prologue)