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Viewing: MA 755 : Introduction to Riemannian Geometry

Last approved: Tue, 30 Aug 2016 08:01:32 GMT

Last edit: Tue, 26 Apr 2016 18:18:36 GMT

Catalog Pages referencing this course
Change Type
Major
MA (Mathematics)
755
014024
Dual-Level Course
Cross-listed Course
No
Introduction to Riemannian Geometry
Intro Riemannian Geometry
College of Sciences
Mathematics (17MA)
Term Offering
Spring Only
Offered Every Year
Spring 2017
Previously taught as Special Topics?
No
 
Course Delivery
Face-to-Face (On Campus)

Grading Method
Graded/Audit
3
16
Contact Hours
(Per Week)
Component TypeContact Hours
Lecture3.0
Course Attribute(s)


If your course includes any of the following competencies, check all that apply.
University Competencies

Course Is Repeatable for Credit
No
 
 
Irina Kogan
Associate Professor
Full

Open when course_delivery = campus OR course_delivery = blended OR course_delivery = flip
Enrollment ComponentPer SemesterPer SectionMultiple Sections?Comments
Lecture1010NoNone
Open when course_delivery = distance OR course_delivery = online OR course_delivery = remote
Prerequisite: MA 555
Is the course required or an elective for a Curriculum?
No
An introduction to smooth manifolds with metric. Topics include: Riemannian metric and generalizations, connections, covariant derivatives, parallel translation, Riemannian (or Levi-Civita) connection, geodesics and distance, curvature tensor, Bianchi identities, Ricci and scalar curvatures, isometric embeddings, Riemannian submanifolds, hypersurfaces, Gauss Bonnet Theorem; applications and connections to other fields.

MA755 is being revised to serve as the second part of a two-course sequence to prepare doctoral students taking the qualifying exam in Differential Geometry (MA 555 is the first course in this sequence). One minor change is being made to the course content in that the choice of applications or special topics at the end of the course is being left to the discretion of the instructor. The course description has been revised to better align with the textbook.  Previously, this sequence consisted of the course MA 518 followed by MA 555.  Overall, the coordinated revision of MA 518, MA 555 and MA 755 serves to strengthen our geometry qualifying exam courses sequence. 


No

Is this a GEP Course?
GEP Categories

Humanities Open when gep_category = HUM
Each course in the Humanities category of the General Education Program will provide instruction and guidance that help students to:
 
 

 
 

 
 

 
 

 
 

 
 

Mathematical Sciences Open when gep_category = MATH
Each course in the Mathematial Sciences category of the General Education Program will provide instruction and guidance that help students to:
 
 

 
 

 
 

 
 

Natural Sciences Open when gep_category = NATSCI
Each course in the Natural Sciences category of the General Education Program will provide instruction and guidance that help students to:
 
 

 
 

 
 

 
 

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Each course in the Social Sciences category of the General Education Program will provide instruction and guidance that help students to:
 
 

 
 

 
 

 
 

 
 

 
 

Interdisciplinary Perspectives Open when gep_category = INTERDISC
Each course in the Interdisciplinary Perspectives category of the General Education Program will provide instruction and guidance that help students to:
 
 

 
 

 
 

 
 

 
 

 
 

 
 

 
 

Visual & Performing Arts Open when gep_category = VPA
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&
 

 
 

 
 

 
 

Global Knowledge Open when gep_category = GLOBAL
Each course in the Global Knowledge category of the General Education Program will provide instruction and guidance that help students to achieve objective #1 plus at least one of objectives 2, 3, and 4:
 
 

 
 

 
Please complete at least 1 of the following student objectives.
 

 
 

 
 

 
 

 
 

 
 

US Diversity Open when gep_category = USDIV
Each course in the US Diversity category of the General Education Program will provide instruction and guidance that help students to achieve at least 2 of the following objectives:
Please complete at least 2 of the following student objectives.
 
 

 
 

 
 

 
 

 
 

 
 

 
 

 
 

Requisites and Scheduling
 
a. If seats are restricted, describe the restrictions being applied.
 

 
b. Is this restriction listed in the course catalog description for the course?
 

 
List all course pre-requisites, co-requisites, and restrictive statements (ex: Jr standing; Chemistry majors only). If none, state none.
 

 
List any discipline specific background or skills that a student is expected to have prior to taking this course. If none, state none. (ex: ability to analyze historical text; prepare a lesson plan)
 

Additional Information
Complete the following 3 questions or attach a syllabus that includes this information. If a 400-level or dual level course, a syllabus is required.
 
Title and author of any required text or publications.
 

 
Major topics to be covered and required readings including laboratory and studio topics.
 

 
List any required field trips, out of class activities, and/or guest speakers.
 

College(s)Contact NameStatement Summary
College of SciencesDavid E Aspnes As a non-mathematician, looks fine to me. By "Divergence Theorem", I assume that you mean the 3D version, since you list Stokes Theorem (a 2D divergence theorem) in the earlier class in the sequence.
College of EngineeringMihail DevetsikiotisECE sees nothing objectionable in these course revisions from their point of view.
No new resources are required due to this course revision. The current version of this course is taught by graduate faculty in our department as part of their regular teaching load.

To learn how to define and compute main geometric  characteristics of a differential manifold with metrics.


Student Learning Outcomes

A student who successfully completes this course will be able to:


1. State  the definition of a metric  and an isometry.


2. Use metrics to define arc-length, distance and volume.


3. Give proofs of and use the Divergence Theorem and Green's Theorem on Riemannian manifolds.


4. State definitions and properties of geometric invariants of Riemannian manifolds: Riemannian curvature, Ricci curvature, scalar curvature.


5. Define and compute Riemannian connection, covariant derivatives, and parallel translation. 


6. Give a definition  of geodesics (as curves with zero acceleration) and write their defining equations.              Prove that geodesics are length-minimizing curves.


7. State and prove Hopf-Rinow completeness theorem.


8. State definitions and properties of geometric invariants of Riemannian submanifolds and hyperserfaces.


9. Distinguish between intrinsic and extrinsic invariants of Riemannian submanifolds. 


10. State the Gauss-Bonnet Theorem. Explain the main ideas behind its proof and its importance as a global-local result


Evaluation MethodWeighting/Points for EachDetails
Homework50Given every 7-10 days
Midterm25One mid-term exam
Final Exam25In-class final exam
TopicTime Devoted to Each TopicActivity
Review of tensors, manifolds, and tensor bundles1 weekLectures
Riemannian, pseudo-Riemannian and sub-Riemannian metrics.1 weekLectures
Connections, covariant derivatives, parallel translation2 weeksLectures
Riemannian (or Levi-Civita) connection, geodesics, normal coordinates2 weeksLectures
Geodesics and distance2 weeksLectures
Curvature tensor, Bianchi identities, Ricci and scalar curvatures2 weeksLectures
Riemannian sub manifolds, hypersurfaces in the Euclidean space2 weeksLectures
The Gauss-Bonnet theorem2 weeksLectures
Applications1 weekLectures
In response to the comment from Physics, the language relating to the Divergence Theorem was made more precise.

mlnosbis 4/15/2016: No overlapping courses. Consultation notes listed above.

ghodge 4/15/2016: Ready for ABGS reviewers. Comment: Assume a standard grading scale. Consider adding grading scale to syllabus

ABGS Reviewer Comments:
--Syllabus does not meet university or Graduate School standards.

ghodge 4/21/2016 return to department and send link to syllabus standards. RESOLVED.
Key: 3539