New User Special Price Expires in

Let's log you in.

Sign in with Facebook


Don't have a StudySoup account? Create one here!


Create a StudySoup account

Be part of our community, it's free to join!

Sign up with Facebook


Create your account
By creating an account you agree to StudySoup's terms and conditions and privacy policy

Already have a StudySoup account? Login here

Geometry of Manifolds

by: Alvena McDermott

Geometry of Manifolds MATH 7350

Marketplace > University of Houston > Mathmatics > MATH 7350 > Geometry of Manifolds
Alvena McDermott
GPA 3.69

Andrei Torok

Almost Ready


These notes were just uploaded, and will be ready to view shortly.

Purchase these notes here, or revisit this page.

Either way, we'll remind you when they're ready :)

Preview These Notes for FREE

Get a free preview of these Notes, just enter your email below.

Unlock Preview
Unlock Preview

Preview these materials now for free

Why put in your email? Get access to more of this material and other relevant free materials for your school

View Preview

About this Document

Andrei Torok
Class Notes
25 ?




Popular in Course

Popular in Mathmatics

This 2 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 7350 at University of Houston taught by Andrei Torok in Fall. Since its upload, it has received 50 views. For similar materials see /class/208387/math-7350-university-of-houston in Mathmatics at University of Houston.


Reviews for Geometry of Manifolds


Report this Material


What is Karma?


Karma is the currency of StudySoup.

You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 09/19/15
Main topics Math 7350 Most items below come with a de nition examples and maybe proof even if not mentioned explicitly Md denotes a manifold of dimension d an 1 Review of multivariable calculus on open subsets of Euclidean spaces 7 higher order derivatives Taylor polynomials 7 the lnverse Function Implicit Function and Rank theorems normal forms for such maps ie the simplest form up to local diffeomorphisms 2 De nition of manifolds with and without boundary We restrict ourselves to second countable Haus dorff spacesi Ckchartsi Maximal atlas differential structurei Topology through charts if the manifold is not already a topological space New charts amp manifolds from old Theorem Whitney Each Ck manifold admits a compatible C00 structure All these C00 structures are smoothly diffeomorphici Thus could assume from now on that all manifolds are smooth 3 Differentiable maps between manifoldsi Critical singular etci points of a map Submersions immersions embeddingsi The lnverse Function Implicit Function and Rank theorems on manifoldsi 4i Submanifoldsi Preimage of a regular point is a submanifoldi Image of an immersion or embedding 5 The tangent space in local coordinates and associated basis through curves or as derivations for smooth manifolds The differential of a map in each of the above representations 6 General vector bundles local trivializations the transition functions the compatibility condition The tangent bundlei Vector elds in various descriptions 7i ODEls in R and on a manifold The ow localglobal oneparameter group associated to a vector eld 8 The Lie derivative of a tensor eld with respect to a vector eld The Lie derivative is actually associated to the oneparameter group determined by the vector eld The bracket of two vector elds the Lie algebra of smooth vector elds 9 Distributions integrability lf X1X2HiXd are commuting vector elds which are linearly independent at each point then locally there is a chart such that Xk i for 1 S k S d The Frobenius theorem the proof in Spivak is more conceptual 0 Tensors alternating forms for a nite dimensional vector space Bases for tensors and forms The wedge producti H CA3 i The bundle of differential forms of a manifold The exterior differential of a form d2 0i Pullback of a form7 its behavior With respect to the wedge product and exterior differential i Closed and exact forms The Poincare Lemmai i Partitions of unity including locally nite covers7 etc i Orientation of a vector space Orientation of a manifold M is orientable iff there exists a nowhere vanishing n formi Orientation induced on the boundary of an oriented manifold i Integration of kforms on singular kchainsi The boundary of a chain 82 0 The Stokes theorem for chainsi i Integration of compactly supported n forms on an oriented n dimensional manifold The Stokes theorem for manifoldsi i Embedding of compact manifolds in RNi The medium Whitney embedding theorem 1936 M lt gt R2n1i


Buy Material

Are you sure you want to buy this material for

25 Karma

Buy Material

BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.


You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

Why people love StudySoup

Steve Martinelli UC Los Angeles

"There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

Jennifer McGill UCSF Med School

"Selling my MCAT study guides and notes has been a great source of side revenue while I'm in school. Some months I'm making over $500! Plus, it makes me happy knowing that I'm helping future med students with their MCAT."

Bentley McCaw University of Florida

"I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"


"Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

Become an Elite Notetaker and start selling your notes online!

Refund Policy


All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email


StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here:

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.