DIFFERENTIAL GEOMETRY

Prerequisites & course information

Requisites: ( MATH 320 , 340 , 341 , 345 , or 375 ) and ( MATH 322 , 341 , 376 , 421 , 467 , or 521 ), graduate/professional standing, or declared in Mathematics VISP (undergraduate or graduate) Course Designation: Core GenEd - Natural Science & Wellness Breadth - Natural Science Level - Advanced L&S Credit - L&S Liberal Arts and Science Grad 50% - Counts toward 50% graduate coursework requirement Repeatable for Credit: No Last Taught: Spring 2026 Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in differential geometry in n-dimensional real space (e.g., vector spaces and inner product spaces, manifolds, Riemannian metric on a manifold, arc length, curvature, torsion, geodesic curves, etc.). Audience: Both Grad & Undergrad 2. Recall and state the standard theorems of differential geometry (e.g., Frenet-Serret equations, Hopf winding number theorem, four vertex theorem for convex curves, etc.), and recall the arguments for these theorems and the underlying logic of their proofs. Audience: Both Grad & Undergrad 3. Compute various quantities related to curves and surfaces in the n-dimensional real space using the definitions and theorems (e.g., the fundamental forms, arc length, curvature, total derivative, etc.). Audience: Both Grad & Undergrad 4. Use the covered theorems in the context of longer arguments by examining their premises (e.g., arguing paths are geodesics, applying the inverse function theorem to show a level set is a manifold, etc. ). Audience: Both Grad & Undergrad 5. Prove or disprove statements related to the above definitions, properties, and theorems using techniques of mathematical argument (direct methods, indirect methods, constructing examples and counterexamples, induction, etc.). Audience: Both Grad & Undergrad 6. Convey arguments using English and appropriate mathematical terminology, notation and grammar. Audience: Both Grad & Undergrad 7. Identify applications of course content in current areas of research. Audience: Graduate Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in differential geometry in n-dimensional real space (e.g., vector spaces and inner product spaces, manifolds, Riemannian metric on a manifold, arc length, curvature, torsion, geodesic curves, etc.). Audience: Both Grad & Undergrad 2. Recall and state the standard theorems of differential geometry (e.g., Frenet-Serret equations, Hopf winding number theorem, four vertex theorem for convex curves, etc.), and recall the arguments for these theorems and the underlying logic of their proofs. Audience: Both Grad & Undergrad 3. Compute various quantities related to curves and surfaces in the n-dimensional real space using the definitions and theorems (e.g., the fundamental forms, arc length, curvature, total derivative, etc.). Audience: Both Grad & Undergrad 4. Use the covered theorems in the context of longer arguments by examining their premises (e.g., arguing paths are geodesics, applying the inverse function theorem to show a level set is a manifold, etc. ). Audience: Both Grad & Undergrad 5. Prove or disprove statements related to the above definitions, properties, and theorems using techniques of mathematical argument (direct methods, indirect methods, constructing examples and counterexamples, induction, etc.). Audience: Both Grad & Undergrad 6. Convey arguments using English and appropriate mathematical terminology, notation and grammar. Audience: Both Grad & Undergrad 7. Identify applications of course content in current areas of research. Audience: Graduate