No /usr/local/cuda/ directory after Cuda installation

I have installed Cuda with this command:

$ sudo apt install nvidia-cuda-toolkit gcc-10

and then checked the version with:

 $ nvcc --version nvcc: NVIDIA (R) Cuda compiler driver Copyright (c) 2005-2021 NVIDIA Corporation Built on Sun_Feb_14_21:12:58_PST_2021 Cuda compliation tools, release 11.2, V11.2.152 Build cuda_11.2.r11.2/compiler.29618528_0

However, there is no /usr/local/cuda/ directory and so I do not know what to do as I think having this directory is necessary to install cuDNN (which is the end goal).

I am using lubuntu 21.04 and an NVIDIA GeForce GT 720M GPU with a legacy 390 driver.

So if you could tell me what to do in order to install Cuda correctly so that there is a /usr/local/cuda/ directory and I can install cuDNN, that would be great!

Edit: After installing cudnn by copying the files from the tar file into /usr/lib/cuda/ (thanks to Tilman) I tried to verify the installation by trying to compile the mnistCUDNN file in the /usr/src/cudnn_samples_v8 directory (as suggested in the Nvidia docs) but that directory also doesn't exist. Any idea on where it would be?

Thank you.

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