On Diffeomorphisms

Differential Geometry
Post Reply
Tsakanikas Nickos
Community Team
Posts: 314
Joined: Tue Nov 10, 2015 8:25 pm

On Diffeomorphisms

#1

Post by Tsakanikas Nickos »

Let $f \ \colon M \longrightarrow N $ and $ g \ \colon N \longrightarrow P $ be smooth maps between smooth manifolds. Show that the composite $ g \circ f $ is smooth. Conclude that the composition of diffeomorphisms is again a diffeomorphism, and show that the converse is not true.

Are there any (reasonable) conditions (imposed on the maps) that guarantee that the converse is also true? The motivation behind this question is the following (for example), coming from Algebraic Geometry: "Let $f \ \colon X \longrightarrow Y$ and $g \ \colon Y \longrightarrow Z $ be morphisms of (noetherian) schemes. If $ g \circ f $ is proper and $g$ is separated, then $f$ is also proper."
Post Reply

Create an account or sign in to join the discussion

You need to be a member in order to post a reply

Create an account

Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute

Register

Sign in

Who is online

Users browsing this forum: No registered users and 10 guests