creating a mesh similar to the image:

Question

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

Answer 1

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

Answer 2

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

Answer 3

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 90^o on the X axis.

Next add an empty and rotate it by 90^o 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.

You should end up with this:

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