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Help me to create a mesh similar to that of the drawing in bold below. Excuse me for my bad drawing skills.

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  • Add a UV sphere (Add>Mesh>UV Sphere)
  • set selection mode to faces :
    enter image description here
  • select the faces you want to keep
  • press Ctrl+I to invert the selection
  • press X, and select delete faces.

enter image description here

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You might want more universal method, depending on what's your goal exactly. Theres a modifier Shrinkwrap that projects one mesh to another. So you can project a mesh with complicated topology to another mesh (very useful for retopology for example).

Add base object (as a base for shape) and a plane (which will be projected on the base shape object): Add a sphere and a plane

Add Shrinkwrap modifier to the object you want to modify (plane in this example) and use sphere as a target object. Project one mesh to another

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Another way this can be done is using two bend modifiers and an empty.

First add a plane, and then add two loop cuts in each direction. In edit mode, rotate it by 90o on the X axis.

Next add an empty and rotate it by 90o on the Y axis

Finally, add two simple deform modifiers and set them to bend. on the second one, change to origin to your empty.

enter image description here

You should end up with this:

enter image description here

This method has an advantage over the other two - it can be adjusted.

enter image description here

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