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)

(Per Week)

Component Type | Contact Hours |
---|---|

Lecture | 3.0 |

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 Component | Per Semester | Per Section | Multiple Sections? | Comments |
---|---|---|---|---|

Lecture | 10 | 10 | No | None |

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

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

Obj. 1) Engage the human experience through the interpretation of culture.

Obj. 2): Become aware of the act of interpretation itself as a critical form of knowing in the humanities.

Obj. 3) Make academic arguments about the human experience using reasons and evidence for supporting those reasons that are appropriate to the humanities.

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

Obj. 1) Improve and refine mathematical problem-solving abilities.

Obj. 2) Develop logical reasoning skills.

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

Obj.O 1) Use the methods and processes of science in testing hypotheses, solving problems and making decisions

Obj. 2) Make inferences from and articulate, scientific concepts, principles, laws, and theories, and apply this knowledge to problem solving.

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

Obj. 1) Examine at least one of the following: human behavior, culture, mental processes, organizational processes, or institutional processes.

Obj. 2) Demonstrate how social scientific methods may be applied to the study of human behavior, culture, mental processes, organizational processes, or institutional processes.

Obj. 3) Use theories or concepts of the social sciences to analyze and explain theoretical and or real-world problems, including the underlying origins of such problems.

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

Obj. 1) Distinguish between the distinct approaches of two or more disciplines.

Obj. 2) Identify and apply authentic connections between two or more disciplines.

Obj. 3) Explore and synthesize the approaches or views of two or more disciplines.

1. Which disciplines will be synthesized, connected, and/or considered in this course?

Each course in the Visual and Performing Arts category of the General Education Program will provide instruction and guidance that help students to:

Obj. 1) Deepen their understanding of aesthetic, cultural, and historical dimensions of artistic traditions.

Obj. 2) Strengthen their ability to interpret and make critical judgements about the arts through the analysis of structure, form, and style of specific works.

Obj. 3) Strengthen their ability to create, recreate, or evaluate art based upon techniques and standards appropriate to the genre.

Each course in the Health and Exercise Studies category of the General Education Program will provide instruction and guidance that help students to:

Obj. 1) Acquire the fundamentals of health-related fitness, encompassing cardio-respiratory and cardiovascular endurance, muscular strength and endurance, muscular flexibility and body composition.

Obj. 2) Apply knowledge of the fundamentals of health-related fitness toward developing, maintaining, and sustaining an active and healthy lifestyle.

Obj. 3) Acquire or enhance the basic motor skills and skill-related competencies, concepts, and strategies used in physical activities and sport.

Obj. 4) Gain a thorough working knowledge, appreciation, and understanding of the spirit and rules, history, safety, and etiquette of physical activities and sport.

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:

Obj. 1) Identify and examine distinguishing characteristics, including ideas, values, images, cultural artifacts, economic structures, technological or scientific developments, and/or attitudes of people in a society or culture outside the United States.

Obj. 2) Compare these distinguishing characteristics between the non-U.S. society and at least one other society.

Obj. 3) Explain how these distinguishing characteristics relate to their cultural and/or historical contexts in the non-U.S. society.

Obj. 4) Explain how these disinguishing characteristics change in response to internal and external pressures on the non-U.S. society.

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:

Obj. 1) Analyze how religious, gender, ethnic, racial, class, sexual orientation, disability, and/or age identities are shaped by cultural and societal influences.

Obj. 2) Categorize and compare historical, social, political, and/or economic processes producing diversity, equality, and structured inequalities in the U.S.

Obj. 3) Interpret and evaluate social actions by religious, gender, ethnic, racial, class, sexual orientation, disability, and/or age groups affecting equality and social justice in the U.S.

Obj. 4) Examine interactions between people from different religious, gender, ethnic, racial, class, sexual orientation, disability, and/or age groups in the U.S.

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)

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 Name | Statement Summary |
---|---|---|

College of Sciences | David 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 Engineering | Mihail Devetsikiotis | ECE 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 Method | Weighting/Points for Each | Details |
---|---|---|

Homework | 50 | Given every 7-10 days |

Midterm | 25 | One mid-term exam |

Final Exam | 25 | In-class final exam |

Topic | Time Devoted to Each Topic | Activity |
---|---|---|

Review of tensors, manifolds, and tensor bundles | 1 week | Lectures |

Riemannian, pseudo-Riemannian and sub-Riemannian metrics. | 1 week | Lectures |

Connections, covariant derivatives, parallel translation | 2 weeks | Lectures |

Riemannian (or Levi-Civita) connection, geodesics, normal coordinates | 2 weeks | Lectures |

Geodesics and distance | 2 weeks | Lectures |

Curvature tensor, Bianchi identities, Ricci and scalar curvatures | 2 weeks | Lectures |

Riemannian sub manifolds, hypersurfaces in the Euclidean space | 2 weeks | Lectures |

The Gauss-Bonnet theorem | 2 weeks | Lectures |

Applications | 1 week | Lectures |

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.

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