Math 565: Differential Geometry - Revision history

Ls5: /* Offered */

2023-04-10T13:25:30Z

‎Offered

Ls5: /* Catalog Information */

2019-11-14T17:08:12Z

‎Catalog Information

Ls5: moved Math 565 to Math 565: Differential Geometry

2013-04-03T22:43:05Z

moved Math 565 to Math 565: Differential Geometry

Cpg: /* Minimal learning outcomes */

2010-07-28T16:31:27Z

‎Minimal learning outcomes

Cpg at 19:00, 4 June 2009

2009-06-04T19:00:37Z

Tlf2 at 23:03, 26 February 2009

2009-02-26T23:03:30Z

Tlf2: /* Description */

2008-08-28T21:32:17Z

‎Description

Tlf2: /* Desired Learning Outcomes */

2008-08-20T22:53:02Z

‎Desired Learning Outcomes

Tlf2: /* Description */

2008-08-20T22:52:37Z

‎Description

Tlf2: New page: == Catalog Information == === Title === Differential Geometry === Credit Hours === 3 === Prerequisite === Math 316. === Description === This course is designed to provide students...

2008-08-20T22:52:08Z

New page: == Catalog Information == === Title === Differential Geometry === Credit Hours === 3 === Prerequisite === Math 316. === Description === This course is designed to provide students...

New page

== Catalog Information ==

=== Title ===
Differential Geometry

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 316]].

=== Description ===

This course is designed to provide students with a rigorous treatment of the theory of differential equations.

== Desired Learning Outcomes ==

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include the differential topology, Riemannian metrics, geodesics, curvature, and integration on manifolds.

=== Prerequisites ===

Students should have taken [[Math 316]] prior to taking this course. Math 316 provides a rigorous background of analysis that is needed to understand many of the proofs in this course.

=== Minimal learning outcomes ===

<div style="-moz-column-count:2; column-count:2;">

</div>

Students should achieve mastery of the topics
below. This means that they should know all relevant definitions,
full statements of the major theorems, and examples of the various
concepts. Further, students should be able to solve non-trivial problems
related to these concepts, and prove theorems in analogy to proofs
given by the instructor.

# Differential topology
#* Differentiable manifolds and smooth maps
#* Tangent space, tangent bundle, derivative of a smooth map
#* Immersions, submersions, and embeddings
#* Orientation
#* Vector fields, brackets
# Riemannian metrics
#* Definition of Riemannian metrics
#* Affine connections
#* Riemannian connections
# Geodesics
#* Definition of geodesics
#* Geodesic flow
#* Minimizing properties of geodesics
#* Exponential map
#* Convex neighborhoods
# Curvature
#* Definitions of curvature, curvature tensor
#* Second fundamental form
#* Sectional and Ricci curvature
#* Jacobi fields
# Integration on manifolds
#* Tensor and vector bundles
#* Exterior algebra
#* Differential forms and exterior derivative
#* Stokes Theorem

=== Additional topics ===

These are at the instructor's discretion as time allows examples are: Hopf-Rinow theorem, spaces of constant curvature, De Rham cohomology, fixed points and intersection numbers, and Morse theory.

=== Courses for which this course is prerequisite ===

None.

[[Category:Courses|565]]

@@ Line 8: / Line 8: @@
 === Offered ===
-On demand (contact department)
+Fall odd years
 === Prerequisite ===

← Older revision		Revision as of 17:08, 14 November 2019
Line 6:		Line 6:
	=== Credit Hours ===		=== Credit Hours ===
	3		3
		+
		+	=== Offered ===
		+	On demand (contact department)

	=== Prerequisite ===		=== Prerequisite ===

← Older revision		Revision as of 23:03, 26 February 2009
Line 66:		Line 66:

	These are at the instructor's discretion as time allows examples are: Hopf-Rinow theorem, spaces of constant curvature, De Rham cohomology, fixed points and intersection numbers, and Morse theory.		These are at the instructor's discretion as time allows examples are: Hopf-Rinow theorem, spaces of constant curvature, De Rham cohomology, fixed points and intersection numbers, and Morse theory.
		+
		+	=== Recommended Texts ===
		+
		+	Riemannian geometry, by Manfredo P. Do Carmo; Differential Geometry and Topology, with a view to dynamical systems, by Keith Burns and Marian Gidea

	=== Courses for which this course is prerequisite ===		=== Courses for which this course is prerequisite ===

@@ Line 12: / Line 12: @@
 === Description ===
-This course is designed to provide students with a rigorous treatment of the theory of differential geometry.
+A rigorous treatment of the theory of differential geometry.
 == Desired Learning Outcomes ==

← Older revision	Revision as of 22:43, 3 April 2013
(No difference)

@@ Line 23: / Line 23: @@
 === Minimal learning outcomes ===
 Students should achieve mastery of the topics
@@ Line 35: / Line 31: @@
 given by the instructor.
 #	Differential topology
 #*	Differentiable manifolds and smooth maps
@@ Line 60: / Line 57: @@
 #*	Exterior algebra
 #*	Differential forms and exterior derivative
-#*	Stokes Theorem
+#*	Stokes Theorem<br><br><br><br><br><br><br>
 === Additional topics ===

@@ Line 8: / Line 8: @@
 === Prerequisite ===
-[[Math 316]].
+[[Math 342]] or equivlaent.
 === Description ===
@@ Line 20: / Line 20: @@
 === Prerequisites ===
-Students should have taken [[Math 316]] prior to taking this course.  Math 316 provides a rigorous background of analysis that is needed to understand many of the proofs in this course.
+Students should have taken [[Math 342]] prior to taking this course.  Math 342 provides a rigorous background of analysis that is needed to understand many of the proofs in this course.
 === Minimal learning outcomes ===

@@ Line 16: / Line 16: @@
 == Desired Learning Outcomes ==
-This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D.  The topics include the differential topology, Riemannian metrics, geodesics, curvature, and integration on manifolds.
+This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D.  The topics include differential topology, Riemannian metrics, geodesics, curvature, and integration on manifolds.
 === Prerequisites ===