Ytukyg

The lady's cat's leg is injured.

Das Bein der Katze der Frau ist verletzt.

Is this the right way to use the german Genitive case to state the above mentioned statement.
Can this sentence be said without using Genitive?
I have studied about genitive case in details but it is very confusing while applying it.

I would really appreciate if someone would explain this along with underlying grammar and other examples, if possible.Thanks a lot in advance.

The sentence you provided is correct, although, as sometimes happens with constructed examples, not very natural.

https://german.stackexchange.com/a/25459/35111 provides an example using three genitives.

Das ist eine Kopie der Partitur des ersten Klavierkonzerts des polnischen Komponisten Chopin

You can easily find examples of multiple genitives when there are deverbal nouns involved.

die Ankündigung der Eröffnung einer neuen Filiale

die Verbesserung der Durchblutung der Extremitäten

die Untersuchung der Entstehung unseres Universums

In standard written German, the genitive can be replaced by von + dative when the dependent noun has no article and is either plural or a mass noun.

die Entlassung von Mitarbeitern

der Verzehr von Fleisch

The double genitive is grammatically correct here, it sounds a bit clumsy, though. Usually one would try to avoid it (if possible, that's not always the case) and use a construction like "Die Katze der Frau hat ein verletztes Bein" (the lady's cat has an injured leg).

I'm afraid that the genitives are not the primary problem, but the fact that both sentences, the English as well as the German one, are unidiomatic. "The leg is injured" sounds very strange in my ears. "Das Bein ist verletzt" is highly unidiomatic and senseless in the final analysis. I consider it wrong.

German ways of expressing that Verletzung are:

Frage: Was ist passiert?

Die Katze hat sich am Bein verletzt.

Die Katze hat sich das Bein gebrochen / verrenkt / verstaucht.

After an X-ray a doctor would say: (Zustandsbescheibung)

Das Bein ist gebrochen / verrenkt / verstaucht. But certainly NOT: *
Das Bein ist verletzt.

If you want to combine verletzt and sein, you can only do so with nouns that denote persons or animals.

Situation: Es hat sich ein Unfall ereignet.

Der Fahrer des Wagens ist verletzt. Die Frau (das Kind; der Mann) ist verletzt. [Der] Fahrer und [der] Beifahrer sind schwer verletzt.
Der Hund ist angefahren worden. Er ist verletzt.

Thus, your lady-cat-leg-sentence could be, with one genitive only:

Die Katze der Frau hat sich am Bein verletzt.

Die Katze der Frau hat sich das Bein gebrochen / verrenkt /
verstaucht.

Das Bein von der Katze der Frau ist gebrochen / verrenkt / verstaucht.

Any language that has possessives will have the problem that double (or more) possessives may sound unwieldy or strange. There is nothing unique about German in that.

I can see four ways of dealing with it:

1. Accept the unwieldy phrase: das Bein der Katze der Frau
   


2. Allow different possessive forms to be mixed: das Bein von der Katze der Frau
   


3. Insist on different possessive forms - ban the repetition of one form


4. Have a special grammatical rule to deal with the problem

(Note that for clarity I am putting everything in the genitive in bold.)

German (and English) go for 1 and 2. I don't know of any that go for 3. The only languages I know that go for 4 are Irish (léamh) and Scots Gaelic. Their 'rule' consists only of not putting anything in the genitive except the last bit. So in Scots Gaelic we would have:

Cas cait na mnà (Bein Katze der Frau) (logical)

Cas cat na mnà (Bein Katze der Frau) (correct)

Unfortunately Katze is the same in nominative and genitive so you have to look carefully at the bold to see the difference! (If you are wondering where the articles have gone, this is in fact the same as the old-fashioned German (der Frau Katze Bein) or the English (the woman's cat's leg) with different word order.)

If anyone knows of any other language that has any special rule for double genitives, please post.