**Course: **MATH 3980T

**Title: **Introduction to Differentiable Manifolds and Lie Groups

**Section: **103 (Class Number 14592) (Tutorial for students in the Honors Tutorial College, HTC)

**Campus: **Ohio University, Athens Campus

**Department: **Mathematics

**Academic Year: **2019 - 2020

**Term: **Spring Semester

**Instructor: **Mark Barsamian

**Contact Information: **My contact information is posted on my web page.

**Office Hours for 2018 - 2019 Spring Semester: **8:45am - 9:30am Mon - Fri in Morton 538

**Course Description: **A previous tutorial on *Differential Geometry* explored curves and surfaces embedded in \( \mathbb{R}^{3} \). In this course, we study abstract *differentiable manifolds*. As with surfaces in the previous tutorial, the definition of a manifold is based on *coordinate neighborhoods*. But there is a fundamental difference between the courses in their approach to differentiation. In the previous course, *differentiation on surfaces* was always obtained by differentiating in the ambient space \( \mathbb{R}^{3} \). This simplified the initial notation, but it actually both complicated and limited the study of the curves and surfaces. In this course, *differentiation on manifolds* is obtained from differentiation in the coordinate neighborhood. The initial notational structure is much more complicated than in the previous tutorial, but it ultimately leads to a much deeper, much richer understanding. Topics will include Manifolds, Lie Groups and Lie Algebras, Vector Fields and Tensor Fields on Manifolds, Integration and Differentiation on Manifolds, and Curvature

**Textbook Information for 2019 - 2020 Spring Semester MATH XXXXT (Barsamian)**

**Title: ***An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised 2 ^{nd} Edition*

**Authors **William M. Boothby

**Publisher: **Academic Press (An imprint of Elsevier Science), 2003

**ISBN: **978-0121160517

**Calendar for 2019 - 2020 Spring Semester MATH 3980T Section 103 (Class Number 14592), taught by Mark Barsamian**

- Week 1: Mon Jan 13 - Fri Jan 17
**Reading**- Boothby Chapter I

**Background Terms:**These come up in Chapter I and are not defined in Chapter I. If you know what these terms mean, explain them to me in a tutorial meeting. If you don’t know what some of them mean, flag them as important terms. You’ll learn about them in your Topology tutorial. (Explain them to me then.)

- The natural topology on a metric space (page 1)
- Vector Space (page 2)
- Inner Product Space (page 2)
- Normed Vector Space (page 3)
- The sequence of inclusions:
- Inner Product Spaces \( \subset \) Normed Vector Spaces \( \subset \) Metric Spaces \( \subset \) Topological Spaces
- Can you give examples of spaces that are in each of these but not in the one to its left?
- Hausdorff Space (page 6)
- Basis for a topology (page 6)
- Subspace topology (page 6)
- Compact set (page 7)
- Locally connected (page 8)
- Locally compact (page 8)
- Normal Space (page 8)
- Metrizable Space (page 8)

**Discuss in Tutorial**- Some of the Background terms from list above
- Subspace topology (page 6)
- Clarify Process of identifying sides of a rectangle, yielding cylinder, mobius band, torus, Klein bottle
- Discuss issue of embedding these spaces in \( \mathbb{R}^3 \) (discussed on page 12). What goes wrong with Klein?
- Discussion of Theorem 4.1 (page 12) mentions theorem classifying compact, orientable surfaces (2-manifolds), citing proof in Massey.
*(Mark B: remember to bring in clearer proof.)* - Study of 3-manifolds, including classification of them, is a rich area of current research.

**Homework 1**from Boothby (Think about all of these. Pick at least 5 to write up and turn in in Week 2.)

- 1.1 # 1, 2, 4, 6
- 1.3 # 1, 3, 4
- 1.4 # 3, 4, 5
- 1.5 # 1, 2, 3, 5, 6 (Ha Ha)

- Weeks 2 & 3: Mon Jan 20 - Fri Jan 31
**Reading:**Boothby Chapter II Sections 1 - 5**Important Concepts:**These come up in Ch 2. (If you have seen any of them before, explain what you know.)

- The classes of functions \( C^k, C^\infty, C^\omega \)
- The set \( C^\infty(a) \), the
*germs of functions at*\( a \). - The distinction between
*linear maps*from \( C^\infty(a) \) to \( C^\infty(a) \) and derivations from \( C^\infty(a) \)to \( C^\infty(a) \).

**Discuss in Tutorial**- A
*vector space*is a set and a field of scalars, with operation of addition of elements of the set and scalar multiplication of elements of the set by the scalars of the field. - For maps from one vector space to another, the only issue is whether the maps preserve the vector space structures of addtion and scalar multiplication. that is, does \( h(ax+by)=ah(x)+bh(y) \)? Maps that do preserve these structures are called
*linear*. - An
*algebra*is a vector space with the extra operation of multiplication \( * \) of elements of the set by each other such that- multiplication is associative: \( x*(y*z)=(x*y)*z \).
- multiplication is distributive: \( x*(y+z)=x*y+x*z \).
- scalar multiplication in the algebra is compatible with multiplication in the field in the following way: \( c(x*y)=(cx)*y=x*(cy) \).

- For maps from one algebra to another, there is of course the issue of whether or not the algebra respects the vector space structure, that is, whether or not the map is linear. But there is also the issue of how the map behaves with respect to the multiplicative structure of the algebra.
- maps that are
*multiplicative*behave this way: \( h(x*y)=h(x)*h(y) \). - maps that are
*derivations*behave this way: \( h(x*y)=h(x)*y+x*h(y) \)

- maps that are
- Example: The set \( C^\infty(U) \) of differentiable functions has function addition and function multiplication. It is an algebra.
- On this algebra, the evaluation map, \( E_a \), defined by \( E_a(f)=f(a) \), where \( f(a) \) denotes the constant function, is a linear map that is multiplicative. That is, \(E_a(f*g)=f(a)g(a)=E_a(f)*E_a (g)\).
- But on this same algebra, the derivative map, \( D \), defined by \( D(f)=\frac{df(x)}{dx} \) is a linear map that is a derivation. That is, \( \frac{d}{dx}(fg)=\frac{df}{dx}*g+f*\frac{dg}{dx} \), the common product rule for derivatives.

**Homework 2 from Boothby Chapter 2 (Turn in on Wed Feb 5)**- 2.1 # 4, 5, 6
- 2.3 # 4
- 2.4 # 3, 5

- Weeks 4 & 5: Mon Feb 3 - Fri Feb 14
**Reading:**- Boothby Chapter II Sections 6, 7 (don't get bogged down in these)
- Boothby Chapter III Sections 1 - 5

**Important Concepts:**- Revisit a couple of manifolds that you saw before ( \( S^2 \), for example) but with Boothby’s precision.
- More sophisticated examples of manifolds not presented as subspaces of R^3.
- Manifolds that can be presented as quotient spaces
- The Projective Space \( P^n(\mathbb{R}) \)
- The Grassman manifolds \( G(k,n) \)

- Immersions, Embeddings, Submanifolds

**Discuss in Tutorial:**- Tuesday, discuss quotient spaces (3.1 – 3.3 topics)
- Thursday, discuss immersions, embeddings, submanifolds (3.4, 3.5 topics)

**Homework 3:**from Boothby Chapter 3 (Turn in on Thu Feb 13)- 3.1 # 2
- 3.2 # 1, 2, 3, 4, 5

**Exam 2**(Assigned Thu Feb 13Turn in on Fri Feb 14)- Boothby 3.1 # 5
- (See definition (2.2) on page 60 of
*open equivalence relation*.) Give an example of a topological space \( X \) and an equivalence relation \( ~ \) on \( X \) such that \( ~ \) is*NOT*open. - 3.3 # 5

- Week 6: Mon Feb 17 - Fri Feb 21
**Reading:**from Boothby Chapter III Differentiable Manifolds and Submanifolds- Section III.4 Rank of a mapping, Immersions
- Section III.5 Submanifolds

**Discuss in Tutorial**- immersions, embeddings, submanifolds (III.4, III.5 topics)

**Homework 4:**from Boothby Chapter III (Turn in on Wed Feb 19)- III.4 # 3, 7
- III.5 # 2, 6, 7

- Week 7: Mon Feb 24 - Fri Feb 28
**Reading:**from Boothby Chapter III Differentiable Manifolds and Submanifolds- Section III.6 Lie Groups
- Section III.7 The Action of a Lie Group on a Manifold

**Homework 5**from Boothby Chapter III (Turn in on Wed Feb 26)- III.6 # 1, 6, 9
- III.7 # 4, 5, 7, 9

- Week 8: Mon Mar 2 - Fri Mar 6
**Reading:**from Boothby Chapter IV Vector Fields on a Manifold- Section IV.1 The Tangent Space at a Point of a Manifold
- Section IV.2 Vector Fields

**Homework 6:**from Boothby Chapter IV (Turn in on Wed Mar 4)- IV.1 # 2, 5
- IV.2 # 2, 5, 6, 7, 8, 10

- Weeks 9,10: Mon Mar 9 - Fri March 20: Spring Break; no meetings
- Weeks 11,12,13: Mon Mar 23 - Fri Apr 10
**Reading:**from Boothby Chapter IV Vector Fields on a Manifold- Section IV.3 One-Parameter and Local One-Parameter Groups Acting on a Manifold
- Section IV.4 The Existence Theorem for Ordinary Differential Equations
- Section IV.5 Some Examples of One-Parameter Groups

**Homework 7:**from Boothby Chapter IV (Turn in on Thu Apr 9)- IV.3 # 2, 5
- IV.4 # 3, 4, 5
- IV.5
- Show that the maps \(F_1, F_2\) in Example 5.10 and map \(F\) in Examples 5.11 are in fact homomorphisms.
- IV.5 # 3, 4, 6

- Week 14: Mon Apr 13 - Fri Apr 17
**Reading:**from Boothby Chapter IV Vector Fields on a Manifold- Section IV.6 One Parameter Subgroups of Lie Groups
- Section IV.7 The Lie Algebra of Vector Fields on a Manifold

**Homework 8**- IV.6 # 1, 2, 6 (Only do these three if you have not previously done them in some other course.)
- IV.6 # 3,4,7
- IV.7 # 2, 3, 4, 6
- IV.7 # 7 or {8,9}

- Week 15: Mon Apr 20 - Fri Apr 24
**Reading:**from Boothby Chapter V Tensors and Vector Fields on Manifolds- V.1 Tangent Covectors
- V.2 Bilinear Forms. The Riemannian Metric
- V.3 Riemannian Manifolds as Metric Spaces

**Homework 9**- V.1 # 2, 3, 5, 7
- V.2 # 2, 4, 9
- V.3 # 1, 2, 3, 4, 7

- Summer: Mon May 25 meeting
**Reading:**from Boothby Chapter V,*Tensors and Vector Fields on Manifolds*, read- Section V.4 Partitions of Unity

**Homework**- Old stuff:
- Find the mistake in
**Remark (5.6)**on page 77 in Section III.5. - What are the dimensions of \( Gl(n,\mathbb{R}),Sl(n,\mathbb{R}),O(n) \) as
*manifolds*? Explain why the numbers make sense. - Review
*Gauss Map*from*Pressley*.

- Find the mistake in
- Boothby V.4
- Exercises # 1, 2, 3, 4
*Section V.4*starts with a remark about non-existence of a non-vanishing \( C^\infty \) vector field on \( S^2 \), and about the associated fact of non-existence of a non-vanishing covector field on \( S^2 \) as well. But by**Theorem (4.5)**, we see that there*is*a \( C^\infty \)*Riemannian Metric*on \( S^2 \). The proof involves the pullback \( \phi^*\psi \) of the usual*inner product*\( \psi \) on \( \mathbb{R}^2 \) using the coordinate maps \( \phi_i \) of a regular covering \( \{U_i,V_i,\phi_i\} \) and an associated*partition of unity*\( \{f_i\} \). What happens if an analogous sort of scheme is attempted to construct a*non-vanishing vector field*on \( S^2 \)? We know something must fail, but what? Think about such a scheme.- We have studied two surfaces that have immersions in \( \mathbb{R}^3 \):
- 2-dimensional real projective space, \(P^2(\mathbb{R})\)
- The Klein Bottle

*Whitney imbedding Theorem*(**Theorem (4.7)**and remarks following) tells us that these can be imbedded in \( \mathbb{R}^4 \). Of course the theorem statement does not give any indication of how to find the imbedding. See if you can find imbeddings for these on the web (probably spelled*embedding*) and present them.

- Summer: Thu May 28 meeting
**Reading:**from Boothby Chapter V,*Tensors and Vector Fields on Manifolds*, read- Section V.5 Tensor Fields
- Section V.6 Multiplication of Tensors

**Homework**- Boothby V.5 Exercises 1, 2, 3, 4, 5, 6
- Boothby V.6 Exercises 1, 2, 3, 4, 5, 6, 7

**Grading**

During the semester, you will accumulate a * Points Total* of up to

**Homework:**Ten @ 20 points each = 200 points possible**Exams:**3 exams @ 200 points each = 600 points possible**Final Exam:**200 points possible

At the end of the semester, your * Points Total* will be converted into your

- 900 - 1000 points = 90% - 100% = A-, A = You mastered all concepts, with no significant gaps.
- 800 - 899 points = 80% - 89.9% = B-, B, B+ = You mastered all essential concepts and many advanced concepts, but have some significant gaps.
- 700 - 799 points = 70% - 79.9% = C-, C, C+ = You mastered most essential concepts and some advanced concepts, but have many significant gaps.
- 600 - 699 points = 60% - 69.9% = D-, D, D+ = You mastered some essential concepts.
- 0 - 599 points = 0% - 59.9% = F = You did not master essential concepts..

**There is no curve.**

**Throughout the semester, your current scores and current course grade will be available in an online gradebook on the Blackboard system.**

page maintained by Mark Barsamian, last updated April 9, 2019