Dear Vishnu M Nair,

Flattening is some transformation (Homeomorphism)

from a space X into another space Y, F: X → Y.

A clear exposition can be found in any introductory article in topology.

As an example: F(circle)= ellipse

F(sphere)=Ellipsoid

etc.

Sometimes we prefer to study the property (P) in Y instead in X.

Where Y ( is the flattened space) characterized by (P) and

to study (P) in Y is better than that in X.

Where the property (P) is preserved under the (Homeomorphism) F.

An excellent application in anatomy and medicine is attached.

https://lspwww.epfl.ch/publications/gigaserver/thesis-saroul.pdf

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4030657/

Best wishes

Thanks. Issam Kaddoura

This is an excellent question. In addition to helpful answer by @ Issam Kaddoura , a bit more can be added.

It has been observed that rank methods (under the name of flattenings) are also in wide use in algebraic geometry for proving tensor rank and symmetric tensor rank lower bounds by

Klim Efremenko, Ankit Garg, Rafael Oliveira, and Avi Wigderson,

Barriers for Rank Methods in Arithmetic Complexity

https://arxiv.org/pdf/1710.09502.pdf

More to the point, it has also been observed by K. Efremenko et al. (p. 5):

The possibly familiar names including partial derivatives, shifted partial derivatives, evaluation dimension, coefficient dimension which are used e.g. in these lower bounds for monotone, non-commutative, homogeneous, multilinear, bounded-depth and other models [Nis91, Smo93, Raz, NW96, Kay12, GKKS14, KLSS14, FSS14, FLMS15, KS14, KS15] are all rank methods....

In other words, a main advantage to flatten two surface shapes is that it facilitates the evaluation of the dimensionality of the flattened shapes. Another important advantage in flattening a pair of shapes (not mentioned by K. Efremenko et al.) is flattening leads to insights into surface structures as well as the more general problem of surface.