heimlich
secretly
In a way that others are not supposed to notice; hidden, secret.
Er hat ihr heimlich einen Liebesbrief in die Tasche gesteckt.
He secretly slipped a love letter into her bag.

secret
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secretly
In a way that others are not supposed to notice; hidden, secret.
Er hat ihr heimlich einen Liebesbrief in die Tasche gesteckt.
He secretly slipped a love letter into her bag.

The word 'heimlich' (secret/secretly) is related to 'das Heim' (home) and 'geheim' (secret). It describes an action that takes place hidden away, as if at home, and is not intended for the public.
Sie hat heimlich das letzte Stück Kuchen gegessen.
She secretly ate the last piece of cake.
'Heimlich' (secret/secretly) can stand before a noun as an adjective or describe a verb as an adverb. As an adverb, it remains unchanged.
Er hatte einen heimlichen Wunsch.
He had a secret wish.
In the meaning of 'verborgen' (hidden), 'heimlich' is often combined with verbs that describe a hidden action: 'heimlich rauchen' (to smoke secretly), 'jemanden heimlich beobachten' (to secretly watch someone), 'sich heimlich treffen' (to meet secretly), or 'heimlich verliebt sein' (to be secretly in love).
Die Kinder haben sich heimlich getroffen, um ihren Plan zu besprechen.
The children met secretly to discuss their plan.
When 'heimlich' describes a feeling, it often appears with verbs such as 'sich freuen' (to be pleased), 'lachen' (to laugh), or 'genießen' (to enjoy). It emphasizes that the joy or satisfaction is internal and not shown openly.
Sie musste heimlich lächeln, als sie an seinen Witz dachte.
She had to smile secretly when she thought of his joke.
open
'offen' and 'heimlich' are opposites here. 'offen' means that you are honest and direct and do not hide anything. 'heimlich' means that something happens in secret and others are not supposed to notice it. So the difference is clear: with 'offen', others may or should know something; with 'heimlich', they specifically should not. This is especially important when it comes to behavior, conversations, or plans.

Für die Prüfung lernt Max heimlich in der Bibliothek, weil er niemandem davon erzählen will.
Max secretly studies for the exam in the library because he does not want to tell anyone about it.
Vor der Reise spricht sie offen mit den Nachbarn über ihre Pläne.
Before the trip, she talks openly with the neighbors about her plans.